Steps in Development of 2 D Turbine Cascades P M V Subbarao Professor Mechanical Engineering Department A Classical Method Recommended by Schlichting.……

Introductory Remarks by H. Schlichting The problem of the flow through cascades is an important one in the whole field of turbines. This is a pure aerodynamic problem. The real development in the flow problems of turbines will be achieved only by a deeper knowledge of complex flow phenomena. This requires extensive theoretical calculations which, however, need careful correlation with experiments.

CASCADE FLOW PROBLEMS :H. Schlichting The very complex cascade flow problem has been split up as follows:- Two-dimensional flow through cascades. –(a) Incompressible and inviscid flow. –(b) Incompressible, viscous flow. –(c) Compressible flow. Three-dimensional flow through cascades. –(a) Secondary flow effects at blade root and blade tip. –(b) Effects due to radial divergence of the blades in cascades of rotational symmetry.

Potential flow Analysis of cascades There are many methods available for potential flow analysis of Cascades A simple method that works for any infinite linear cascade composed of arbitrary shaped blades separated by a uniform spacing h is called as Single conformal Transformation method. This method solves the problem, irrespective of the blade shape, spacing or stagger.

The linear Cascade of Infinite number of blades This physical plane that contains this geometry is called as the Z 1 plane.

Single conformal Transformation method It is well known that the potential flow past an isolated airfoil can be easily solved using high order vortex panel methods. The idea of this is to directly utilize the standard higher order vortex panel method to solve the cascade problem by after utilizing a conformal mapping. This method transforms the infinite cascade to a single closed contour. This avoids the series of transformations that are usually employed in pure conformal transformation methods in order to reduce this single closed contour to a regular circle. This method can use the panel code of the isolated airfoil problem to solve the cascade problem.

Conformal transformation For the single conformal transformation, the mapping function is: where h is the spacing between blades in the linear-cascade in the physical Z 1 -plane. Once this transformation is applied, all the blades in the linear cascade system shown in the Z 1 -plane are transformed into a single contour in the Z 2 -plane.

Infinite number of Blades in Z 2 -- Plane The far upstream and downstream velocities are represented in this plane by : the point source/sink of strength Q & the point vortices (  upstream,  downstream ) at the locations indicated by the filled in circles.

The Capacity of Cascade The point source of strength Q = V 1 h cos(α 1 ) represents the flow rate between two blades that is dependent on the horizontal velocity component (V 1 cos α 1 ). The vertical velocity component (V 1 sin α 1 ) is represented by a point vortex.  upstream = V 1 h sin(α 1 ), where V 1 and α 1 are the inlet velocity and angle of inclination far upstream of the cascade. Similarly, the exit conditions (at −∞) in the Z 1 -plane can be represented by a combination of a sink (−Q) and a vortex (  downstream ) located at (−1, 0) in the Z 2 -plane. The mass conservation in the Z 1 -plane requires the sink magnitude to be the same as that of the source, while the downstream vertical velocity component, represented by the point vortex (  downstream ) in the Z 2 -plane, will change from the upstream value.

The potential flow around the single closed contour in the Z 2 -plane in the presence of the source (Q) and vortex (  upstream ) at (+1, 0) and the sink ( − Q) and the unknown vortex (  downstream ) at ( − 1, 0) is solved using the vortex panel method. The Method

Vortex Panel Method for Design of Cascade Discretize the transformed single contour in to m vortex panels with m + 1 nodes. The vorticity distribution along each panel is linear as shown in the inset.

Method of Solution In this panel method, the number of unknowns in the form of nodal vorticity strengths is equal to one more than the number of panels (m). In this problem, the vorticity at the sink location (downstream) is an additional unknown. Therefore, there are (m+2) unknowns and the same number of equations are required to solve for the unknowns. The impermeability condition at the control points on the panels and the Kutta condition at the trailing edge yield (m+1) equations. The extra (m + 2) equation is given by the fact that in the z1-plane, the change in vertical velocity from far upstream to far downstream of the cascade should be equal to the circulation around one blade.

m+2 Condition

The Potential Velocity in Z Plane

The Potential Velocity in Z Plane

The Cause Created by a Cascade stagger (β) = 37·5◦, solidity (σ ) = 1·01 and α 1 = 53·5◦.

Comparison with Experimental Results

Optimization of Shape for Better Cause

Configurations of baseline and optimum cascades

Shapes of baseline and optimum cascades

Losses in 2D Cascades For two-dimensional cascades, the main object of the investigations has been to find a way to calculate theoretically the loss coefficients of the cascade. They depend on the geometrical and aerodynamic parameters of the cascade. A turbulent boundary layer computation has been incorporated into the inviscid design. This method solves three ordinary differential equations for three independent parameters, the momentum thickness, the shape factor and the entrainment coefficient.

Cascade Boundary Layer