CN104537234A - One-dimensional high-low-pressure turbine transition flow channel optimization design method - Google Patents

Abstract

The invention discloses a one-dimensional optimal design method for a high-low pressure turbine transition channel, which is used to solve the technical problem of poor practicability of the existing method. The technical solution is to use Bezier curves to construct the end wall profile equation that satisfies the geometric constraints of the transition channel. Then generate disturbances satisfying the orthogonal polynomials, and correct the one that is too large in order to prevent the flow channel from being deformed after the disturbance. And based on the existing transition channel geometry line equation, a new high and low pressure turbine transition channel profile equation is generated. Then, the flow field performance parameters are obtained by solving the flow control equation on the middle line of the transition flow channel of the high and low pressure turbine. Taking the total pressure recovery coefficient as the optimization objective function, the composite method is adopted until the iterative results meet the accuracy requirements. The invention uses the Bezier curve as the basis to construct the profile equation of the high-low pressure turbine transition channel, and optimizes the high-low pressure turbine transition channel. The method overcomes the problem that the existing method is not suitable for the rapidly expanding flow channel with a relatively large expansion angle, and has strong practicability.

Description

One-dimensional optimization design method of high and low pressure turbine transition channel

technical field

The invention relates to a design method for a high-low pressure turbine transition channel, in particular to a one-dimensional optimization design method for a high-low pressure turbine transition channel.

Background technique

The turbine transition channel connecting the high-pressure turbine and the low-pressure turbine is an important part of the large bypass gas turbine engine, and the air flow in the turbine transition channel needs to undergo a deceleration and diffusion process. According to the classic diffuser theory, the air flow is easily separated in the diffuser channel. In order to avoid the flow loss caused by the separation, the axial length of the channel must be increased to reduce the degree of diffusion, and the increase in the axial length will inevitably bring about the increase in the weight and cost of the turbine. Increase.

For a long time, the design of the turbine transition passage has not received enough attention, and the design method has stopped at the classic diffuser theory. However, with the development of aero-engine technology, the size of the core engine has been continuously reduced, and the temperature tolerance of the turbine material has been continuously increased. Turbine cooling technology has developed rapidly, and the classic transition channel design method can no longer meet the needs of current aero-engine design.

The document "Optimum Design of Turbine Transition Flow Passage Based on One-Dimensional Model "Propulsion Technology", Issue 2, 2012, Pages 179-184" discloses a performance prediction method for one-dimensional turbine transition flow passage. In this method, the geometrical conditions of the design should be clarified first, and the initial design of the transition flow channel should be completed according to the known conditions; then, based on the initially designed flow channel, the perturbation basis function satisfying the orthogonal condition should be given to perturb the original design curve; Then, the weighted average of the pressure recovery coefficient and the total pressure recovery coefficient is used as the objective function, and the simplex method is used for optimization, and iterated several times until the required accuracy is achieved, then the one-dimensional optimization design of the transition channel is completed.

The existing method cannot be used for the design of the rapidly expanding flow channel with a large expansion angle, and the applicable range is narrow.

Contents of the invention

In order to overcome the disadvantage of poor practicability of the existing methods, the present invention provides a method for one-dimensional optimal design of transition flow channels of high and low pressure turbines. In this method, the Bezier curve is used to construct the end wall profile equation that satisfies the geometric constraints of the transition channel. Then generate disturbances satisfying the orthogonal polynomials, and correct the one that is too large in order to prevent the flow channel from being deformed after the disturbance. And based on the existing transition channel geometry line equation, a new high and low pressure turbine transition channel profile equation is generated. Then, the flow field performance parameters are solved from the flow control equation on the midline of the high-low pressure turbine transition channel. Finally, the total pressure recovery coefficient is used as the optimization objective function, and the composite method is used until the iterative results meet the accuracy requirements. The invention uses the Bezier curve as the basis to construct the profile equation of the high-low pressure turbine transition channel, and adopts the compound shape method with constraints to optimize the high-low pressure turbine transition channel. The method overcomes the problem that the method in the background technology is not suitable for the rapidly expanding flow channel with a relatively large expansion angle, and has strong practicability.

The technical solution adopted by the present invention to solve the technical problem is: a one-dimensional optimal design method for the high-low pressure turbine transition channel, which is characterized in that the following steps are adopted:

According to the size and geometry of the inlet and outlet of the known flow channel, calculate θ _c and determine L _m according to the formula (1). θ _c is the equivalent conical expansion angle; L _m is the meridian length of the transition channel; A ₁ is the normal area of the transition channel inlet; A ₂ is the normal area of the transition channel outlet.

${\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\sqrt{{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}}{{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\sqrt{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

The casing and hub profile of the turbine transition channel are respectively constructed by two 4th-order Bezier curves with 5 control points, and the parametric equation of each curve is formula (2). is the Bernstein polynomial; m is the order of the polynomial; t is the control parameter of the curve, and its value ranges from 0 to 1; N _i is the coordinate of the control point.

$\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

The radius and slope of the inlet and outlet centerlines of the flow channel are known from the geometric constraints, and the value of the second-order derivative of the preset centerline is 0 at the inlet and outlet, and the initial design equation of the curve is obtained.

The perturbation basis functions use orthogonal polynomials satisfying the following conditions. i is the order of the basis function; L is the axial length of the transition channel.

When x=0.0 or L, $P_i\left(x\right)=\frac{{\mathrm{dP}}_i}{\mathrm{dx}}=\frac{d^2{P}_i}{{\mathrm{dx}}^2}=0---\left(3\right)$

The first basis function is the lowest-order polynomial that satisfies formula (3), and the higher-order basis functions should satisfy both formula (3) and formula (4).

${<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dx</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&NotEqual;</span>\mathrm{<span\; class="notranslate">\; j</span>}\\ <span\; class="notranslate">\; \&NotEqual;</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; j</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

The perturbation is the linear superposition of the orthogonal polynomial basis functions.

Compare the resulting perturbation maximum with 0.05h(0). If the maximum value of the disturbance is greater than 0.05h(0), then reduce the obtained disturbance to the maximum value equal to 0.05h(0). h(0) is the height of the runner inlet.

Dimensionlessization is performed with the function 2 norm shown in Equation (5).

P ₁ (x)=x ³ (Lx) ³

${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

The reference base of the disturbance is the current turbine transition channel profile.

Solve the formulas (6), (7), (8) and (9), that is, the one-dimensional flow control equation, and solve the performance parameters of the flow channel.

$\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;rb\&rho;</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; b</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\frac{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; r</span>}{\mathrm{<span\; class="notranslate">\; CC</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

$\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}\frac{\mathrm{<span\; class="notranslate">\; dp</span>}}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&phi;</span>}}{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\frac{{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; CC</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}}{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; dI</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; C</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; C</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

In the formula, r is the radius of the centerline of the flow channel; b is the width of the flow channel perpendicular to the center line; C _m is the meridional velocity; ρ is the density; B is the blockage factor considering the influence of the boundary layer and separation; C is the mass flow rate; C is the full velocity; m is the direction of the meridian streamline; p is the static pressure; C _θ is the tangential velocity; φ is the angle between the meridian streamline and the axis; c _f is the surface friction coefficient; IC is the curvature loss term; ID is the diffusion loss term; H is the total enthalpy; h is the static enthalpy.

The optimization objective function is formula 10. σ is the total pressure recovery coefficient.

maxobj=σ (10)

The new surface equation is obtained by optimizing with the complex shape method with constraints.

The approximation distance of each vertex of the composite shape satisfies the given accuracy as the standard to judge whether it is converged. If it converges, the optimization will end. If it does not converge, a new disturbance will be generated. Repeat the self-generated disturbance basis function to start a new optimization process.

The beneficial effect of the invention is that: the method adopts the Bezier curve to construct the end wall profile line equation satisfying the geometric constraints of the transition channel. Then generate disturbances satisfying the orthogonal polynomials, and correct the one that is too large in order to prevent the flow channel from being deformed after the disturbance. And based on the existing transition channel geometry line equation, a new high and low pressure turbine transition channel profile equation is generated. Then, the flow field performance parameters are solved from the flow control equation on the midline of the high-low pressure turbine transition channel. Finally, the total pressure recovery coefficient is used as the optimization objective function, and the composite method is used until the iterative results meet the accuracy requirements. The invention uses the Bezier curve as the basis to construct the profile equation of the high-low pressure turbine transition channel, and adopts the compound shape method with constraints to optimize the high-low pressure turbine transition channel. The method overcomes the problem that the method in the background technology is not suitable for the rapidly expanding flow channel with a relatively large expansion angle, and has strong practicability.

The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments.

Description of drawings

Fig. 1 is a flow chart of the one-dimensional optimal design method for the high-low pressure turbine transition channel of the present invention.

Fig. 2 is the flow path profile diagram that the method of the present invention obtains, and the dotted line is before optimization, and the solid line is after optimization.

Detailed ways

Refer to Figure 1-2. The specific steps of the one-dimensional optimal design method for the high-low pressure turbine transition channel of the present invention are as follows:

According to the known size and geometry of the inlet and outlet of the runner, the initial constraints given in this example are as follows:

Then calculate θ _c according to formula (1) and determine L _m . According to Formula 1, the equivalent expansion angle θ _c =17.6895° is obtained, and the flow channel is a rapidly expanding flow channel.

θ _c is the equivalent conical expansion angle; L _m is the meridian length of the transition channel; A ₁ is the normal area of the transition channel inlet; A ₂ is the normal area of the transition channel outlet.

${\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\sqrt{{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}}{{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\sqrt{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

The casing and hub profile of the turbine transition channel are respectively constructed by two 4th-order Bezier curves with 5 control points, and the parametric equation of each curve is formula (2). is the Bernstein polynomial; m is the order of the polynomial; t is the control parameter of the curve, and its value ranges from 0 to 1; N _i is the coordinate of the control point.

$\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

The radius and slope of the inlet and outlet centerlines of the flow channel are known from the geometric constraints, and the value of the second-order derivative of the preset centerline is 0 at the inlet and outlet, and the initial design equation of the curve is obtained.

The perturbation basis functions use orthogonal polynomials satisfying the following conditions. i is the order of the basis function; L is the axial length of the transition channel.

When x=0.0 or L, $P_i\left(x\right)=\frac{{\mathrm{dP}}_i}{\mathrm{dx}}=\frac{d^2{P}_i}{{\mathrm{dx}}^2}=0---\left(3\right)$

The first basis function is the lowest-order polynomial that satisfies formula (3), and the higher-order basis functions should satisfy both formula (3) and formula (4).

${<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dx</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&NotEqual;</span>\mathrm{<span\; class="notranslate">\; j</span>}\\ <span\; class="notranslate">\; \&NotEqual;</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; j</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

The perturbation is the linear superposition of the orthogonal polynomial basis functions.

Compare the resulting perturbation maximum with 0.05h(0). If the maximum value of the disturbance is greater than 0.05h(0), then reduce the obtained disturbance to the maximum value equal to 0.05h(0). h(0) is the height of the runner inlet.

Dimensionlessization is performed with the function 2 norm shown in Equation (5).

P ₁ (x) ₌ x ³ (Lx) ³

${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

The reference base of the disturbance is the current turbine transition channel profile.

Solve the formulas (6), (7), (8) and (9), that is, the one-dimensional flow control equation, and solve the performance parameters of the flow channel.

$\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;rb\&rho;</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; b</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\frac{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; r</span>}{\mathrm{<span\; class="notranslate">\; CC</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

$\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}\frac{\mathrm{<span\; class="notranslate">\; dp</span>}}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&phi;</span>}}{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\frac{{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; CC</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}}{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; iGO</span>}}_{\mathrm{<span\; class="notranslate">\; D.</span>}}}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; C</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

$\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; C</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

In the formula, r is the radius of the centerline of the flow channel; b is the width of the flow channel perpendicular to the center line; C _m is the meridional velocity; ρ is the density; B is the blockage factor considering the influence of the boundary layer and separation; C is the mass flow rate; C is the full velocity; m is the direction of the meridian streamline; p is the static pressure; C _θ is the tangential velocity; φ is the angle between the meridian streamline and the axis; c _f is the surface friction coefficient; IC is the curvature loss term; ID is the diffusion loss term; H is the total enthalpy; h is the static enthalpy.

The optimization objective function is formula 10. σ is the total pressure recovery coefficient.

maxobj=σ (10)

The new surface equation is obtained by optimizing with the complex shape method with constraints.

The approximation distance of each vertex of the composite shape satisfies the given accuracy as the standard to judge whether it is converged. If it converges, the optimization will end. If it does not converge, a new disturbance will be generated. Repeat the self-generated disturbance basis function to start a new optimization process.

In this embodiment, the aerodynamic parameters for solving the control equation are given, as shown in the following table:

Referring to Fig. 1, after 273 steps of iteration, the optimized flow channel shape line is achieved to meet the precision requirements.

Referring to Figure 2, we can see the comparison diagram before and after the optimization of the flow channel. The dotted line in the figure is the profile line of the high-low pressure turbine transition channel before optimization, and the solid line is the profile line of the high-low pressure turbine transition channel after optimization.

In this example, a rapidly expanding transition channel is calculated. Before optimization, the pressure recovery coefficient Cp is 0.561, and the total pressure loss coefficient ω is 0.0135. After optimization, the pressure recovery coefficient Cp is 0.585, and the total pressure loss coefficient is 0.0109. The coefficient is reduced by 19.3%. .

It can be seen from this example that the optimization effect of this method is still good for the one-dimensional rapid expansion channel that cannot be optimized by the existing methods.

Claims (1)

1. A high and low pressure turbine transition channel one-dimensional optimal design method is characterized in that comprising the following steps: According to the size and geometry of the inlet and outlet of the known runner, calculate θ _c and determine L _m according to the formula (1); θ _c is the equivalent conical expansion angle; L _m is the meridional length of the transition channel; A ₁ is the transition channel inlet method to the area; A ₂ is the normal area of the outlet of the transition channel; ${\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>\frac{\sqrt{{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\sqrt{{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}}{{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\sqrt{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ The casing and hub profile of the turbine transition channel are respectively constructed by two 4th-order Bezier curves with 5 control points, and the parametric equation of each curve is formula (2); is the Bernstein polynomial; m is the order of the polynomial; t is the control parameter of the curve, and its value ranges from 0 to 1; N _i is the coordinate of the control point; $\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ The radius and slope of the inlet and outlet centerlines of the flow channel are known from the geometric constraints, and the value of the second derivative of the preset centerline is 0 at the inlet and outlet, so as to obtain the initial design equation of the curve; The disturbance basis function adopts an orthogonal polynomial satisfying the following conditions; i is the order of the basis function; L is the axial length of the transition channel; When x=0.0 or L, $P_i\left(x\right)=\frac{d{P}_i}{\mathrm{dx}}=\frac{d^2{P}_i}{d{x}^2}=0---\left(3\right)$ The first basis function is the lowest-order polynomial that satisfies formula (3), and higher-order basis functions should satisfy both formula (3) and formula (4); ${<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dx</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&NotEqual;</span>\mathrm{<span\; class="notranslate">\; j</span>}\\ <span\; class="notranslate">\; \&NotEqual;</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; j</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ The perturbation is the linear superposition of the orthogonal polynomial basis functions; Compare the maximum value of the obtained disturbance with 0.05h(0); if the maximum value of the disturbance is greater than 0.05h(0), reduce the obtained disturbance to the maximum value equal to 0.05h(0); h(0) is the inlet of the flow channel high; Carry out dimensionless with the function 2 norm shown in formula (5); P ₁ (x)=x ³ (Lx) ³ ${\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ The reference datum of the disturbance is the current turbine transition channel profile; Solve the formulas (6), (7), (8) and (9), that is, the one-dimensional flow control equation, and solve the performance parameters of the flow channel; $\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;rb\&rho;</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\stackrel{<span\; class="notranslate">\; .</span>}{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$ $\mathrm{<span\; class="notranslate">\; b</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\frac{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; rC</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$ $\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}\frac{\mathrm{<span\; class="notranslate">\; dp</span>}}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&phi;</span>}}{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\frac{\mathrm{<span\; class="notranslate">\; d</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; C</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}}{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; d</span>}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}{\mathrm{<span\; class="notranslate">\; dm</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; C</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$ $\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; C</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$ In the formula, r is the radius of the centerline of the flow channel; b is the width of the flow channel perpendicular to the center line; C _m is the meridional velocity; ρ is the density; B is the blockage factor considering the influence of the boundary layer and separation; C is the mass flow rate; C is the full velocity; m is the direction of the meridian streamline; p is the static pressure; C _θ is the tangential velocity; φ is the angle between the meridian streamline and the axis; c _f is the surface friction coefficient; IC is the curvature loss term; ID is the diffusion loss term; H is the total enthalpy; h is the static enthalpy; The optimization objective function is formula 10; σ is the total pressure recovery coefficient; max obj = σ (10) A new surface equation is obtained by optimizing the compound shape method with constraints; The approximation distance of each vertex of the composite shape satisfies the given accuracy as the standard to judge whether it is converged. If it converges, the optimization will end. If it does not converge, a new disturbance will be generated. Repeat the self-generated disturbance basis function to start a new optimization process.