of Compressors and Turbines (AE 651) Autumn Semester 2009 - - PowerPoint PPT Presentation

Aerodynamics

• f

Compressors and Turbines

(AE 651)

Autumn Semester 2009

Instructor : Bhaskar Roy Professor, Aerospace Engineering Department I.I.T., Bombay e-mail : aeroyia@aero.iitb.ac.in

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Fundamental of Fluid mechanics are often expressed mathematically as Partial Differential Equations (PDEs), mostly of second order PDEs. Generally the governing equations are a set of coupled, non-linear PDEs valid within an arbitrary (or irregular) domain and are subject to various initial and boundary conditions. Analytical solutions of various fluid mechanic equations are limited. This is mainly due to imposition of various boundary conditions of typical fluid flow problems. Experimental fluid mechanics provides some fluid flow

• information. Hard ware and instruments often limit the

extent and details of information available. Expts are

• ften used for validation of CFD solutions. Together they

(data produced) are used for design purposes.

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Linear and Non-linear PDEs

∂u ∂u = -a ∂t ∂x

Linear : (Wave Equation)

∂u ∂u = -u ∂t ∂x

Non- Linear (Inviscid Flow – Burgess eqn) 2nd Order equation

. A B C D x y x x y E F G y                        

2 2 2 2 2 2 2

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Assume that    (x,y) is a solution of the diffl eqn This solution, typically is a surface in space, and the solutions produce space curves, called charateristics. 2nd order derivatives along the characteristics are often indeterminate and may be discontinuous across the

• characteristics. The 1st order derivatives are continuous.

A simpler version of the 2nd order equation may be written as:

           

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dy dy A

• B

+C = dx dx

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Solution of this yields the equations of the characteristics in physical space :

     

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dy B± B - 4AC = dx 2A

These characteristic curves can be real or imaginary depending on the values of (B2 – 4AC). A 2nd order PDE is classified according to the sign of : (a) (B2 – 4AC) < 0 ----- Elliptic - M<1.0 – Subsonic flow (b) (B2 – 4AC) = 0 ----- Parabolic M =1.0 – Sonic flow (c) (B2 – 4AC) > 0 ----- Hyperbolic M>1.0 –Supersonic flow

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades When a real solution exists the zone of influence (downstream) is

• finite. Similarly, the

zone of dependence (upstream) is also finite. Zone of Influence (horizontal shades) Zone of Dependence (vertical shades)

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Elliptic equations

• r,

= ( , ) x y f x y        

2 2 2 2

The domain of solution for elliptic PDE is a closed region. BCs provide the solution within the domain The domain of solution for an elliptic PDE

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

Parabolic equation

T T t x       2 2

The solution domain is normally an open

• region. Parabolic PDE

has one characteristic

• line. One IC and two

BCs are rqeuired for complete solution. The Domain of Solution of a Parabolic PDE CFD of Turbomachinery Blades

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Hyperbolic Equations

t x        

2

2 2 2 2

t x         

1st Order HE – One IC is required 2nd Order HE – 2 ICs and 2 BCs Reqd. Solution of HEs for supersonic flow have often been done with Method of Characteristics with two independent

• variables. Along the Characteristic line PDE reduces to ODE

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Initial and Boundary conditions (supplementary condns) ICs : A dependant variable is specified at some initial condn BCs : A dependent variable or its derivative must satisfy on the boundary of the domain of the PDE 1) Dirichlet BC : Dependent variable prescribed at boundary 2) Neumann BC: Normal gradient of the d.v. is specified 3) Robin BC : A linear combination of Dirichlet & Neumann 4) Mixed BC : Some part of the boundary has Dirichlet bc and some other part has Neumann bc Body Surface Far Field Symmetry In / Outflow BCs

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Grid generation PDEs Algebraic equations : Finite Difference Equations Various Finite Difference Techniques Computational space

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Structured Grid generation Domain Transformation Orthogonal Grid Grid without Orthogonality

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades

Unstructured Grid generation

Domain Nodalization => Triangulation

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Through Flow Blade Section Design Blade section stacking Three-Dimensional Flow Analysis Blade-to-Blade design Blade-to-Blade Analysis Blade Construction Full Blade Structural and Aero-elastic analysis Blade design system

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Through Flow Program Input : a) annulus Information Blade row exit information Inlet profiles of Pr, Temp, 1 Inlet Mass flow Rotational speeds of rotors Blade geometry, Loss distributions Passage averaged perturbation terms Output : b) Blade row inlet and exit conditions Streamline definition and streamtube height

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades

Through Flow Program

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades

Through Flow Program

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Blade-to-Blade program Input : Blade geometry Inlet and Exit Velocity distribution Streamline Definition Output : Surface velocity distribution Profile and loss distribution Section Stacking Program Input : Blade section geometry Stacking points and stacking line Axial and Tangential leans (sweep and Dihedral) Output : Three-Dimensional blade geometry

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades 2D MISES code for Cascade Analysis Blade-to-Blade program

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades 2D MISES code for Cascade Analysis Blade-to-Blade program

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Fluent

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades Fluent

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades CFX-Ansys

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades CFX-Ansys

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades CFX-Ansys

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades CFX-Ansys

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades

Vector diagram of tip flows with (a) Rotating Frame

CFX-Ansys

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AE 651 - Prof Bhaskar Roy, IITB

Lecture-20

CFD of Turbomachinery Blades

Vector diagram of tip flows with (b) Stationary frame,

CFX-Ansys

AE 651 - Prof Bhaskar Roy, Aerospace Engg. Dept., IIT,Bombay

Thank you very much for participating in this course AE 651