37464 - Turbomachines M

Learning outcomes

The student masters the basic methodologies to deal with the thermo-fluid dynamic design of turbomachinery. The logical path adopted allows the student to acquire knowledge for the sizing of machines, through the use of statistical correlations, the one-dimensional sizing and the two-dimensional approach, with the development and application of aerodynamic techniques, considering driving and operating machines with incompressible and compressible fluid.

Course contents

RECALL OF TENSORIAL ALGEBRA First and second order tensors. Operations between tensors: internal product, external product. Power of a Tensor. Principal invariants, eigenvalues and eigenvectors. Rotation tensor in Cartesian coordinates.

CURVILINEAR COORDINATES Determination of the gradient, divergence, rotor and Laplacian in orthogonal Cartesian, cylindrical and axisymmetric coordinates.

ELEMENTS OF CONTINUOUS MECHANICS Deformation gradient tensor. Velocity vector gradient. Time derivative of the deformation tensor. Divergence of a tensor. The stress tensor, its symmetry and Cauchy's Th. Material or Lagrangian derivative of a scalar and a vector.

EQUATIONS OF MOTION Ccontinuity, momentum and energy, Crocco formula for a compressible fluid in a three-dimensional space and in Cartesian and cylindrical coordinates.

CONSTITUTIVE EQUATION FOR THE STRESS TENSOR Navier Sotkes equations. FLOW FIELDS Vorticity and irrotational flow fields. The vector potential and its properties. Solenoidal motion field: the velocity potential and its properties. Orthogonality of iso potential lines of velocity iso potential vector.

COMPLEX VARIABLE FUNCTIONS Cauchy Reiemann conditions and holomorphic functions. The complex potential and the complex velocity. The conformal transformation and maintenance of the angles between the tangents to the curves in a plane and the corresponding tangents to the curves in the transformed plane. Complex potential in the case of a point source and in the case of a free vortex source.

NACA PROFILES Classification. Construction of a 4-digit NACA profile. Profile libraries present in literature and online (airfoiltools).

FORCE ON AN ISOLATED PROFILE Lift, lift and pressure coefficient for a profile. Using the complex force to determine the Blasius formula for an isolated profile. Correlation between lift and circulation for an isolated profile. Th. of Joukowsky. Kutta condition. Determination of the lift and resistance coefficient for a profile placed in an array in the case of absence and presence of leaks. Analytical expression for the lift and drag coefficients for a moving profile.

APPROXIMATION OF THREE-DIMENSIONAL FLOW IN TURBOMACHINES Axisymmetric flows. Orthogonal axisymmetric coordinates. Stokes current function. Expression of the Stokes current function on the S2 surface (meridian plane) and in the case of the S1 surface blade to blade. Flow analysis in a blade channel using dedicated software (Mathematica, Matlab, Fluent, OpenFoam).

ASPECT RATIO, SOLIDITY, INCIDENCE AND DEVIATION IN AXIAL COMPRESSOR RANGE Correlations of Howell, Carter, Lieblein, Weinig. De Haller verification. The methods of Zweifel and McKenzie. Use of statistical data from literature: Mellor curves.

TRANSONIC COMPRESSORS Normal and oblique shock waves. Shock wave attached to the profile and detached wave (Bow wave) in a blade channel of a transonic compressor, energy losses.

INFLUENCE OF REYNOLDS NUMBER ON THE PERFORMANCE OF BLADE PROFILES Use of Prandtl theory for the determination of the drag coefficient in the cascade in the case of laminar and turbulent flow and co-presence of laminar and turbulent flow. Influence of the Mach number on the performance of blade profiles.

ENERGY LOSSES IN THE LAYERS OF PALE PROFILES Losses due to the deviation defect for axial (Konig theory) and radial (Busemann theory) profiles. Losses due to passing vortices (hub and tip endwall losses); to tip vortices (tip clearance losses), to horseshoe vortices (horsesohoe losses).

DESIGN and DESIGN OF A TURBOMACHINE Radial blades; determination of the profile of a radial blade by conformal transformation. Use of CAD software for 3D pallet tracking. Axial blades; Simple radial equilibrium. Types of blade design with the free vortex design method (work, degree of reaction, alpha and beta angles). Circumferential velocity trend with Constant reaction design and Exponential design. Comparison between methods.

A video presentation of the teaching is available through the link at the top right of this page.

Readings/Bibliography

Lecture notes provided by the teacher.

Teaching methods

Explanation in the classroom using an electronic blackboard. I use turbomachinery modeling software.

Assessment methods

The student will have to face a final oral test on the developed program.

Teaching tools

Using software for flow and blade analysis in the turbomachinery in both 2D and 3D.
Parts of Turbines and Compressors brought to class and explained to students.

Links to further information

https://youtu.be/ApgdrFq4KHM

Office hours

See the website of Antonio Peretto