HYDRODYNAMIC AND THERMAL TWO DIMENSIONAL BOUNDARY LAYERS DEVELOPMENT BETWEEN ROTATING TURBINE BLADES

The hydrodynamic and thermal boundary layers have great effect on the fluid flow and heat transfer between rotating turbine blade. In the present work, the flow and heat transfer is analyzed numerically by solving two dimensional incompressible boundary layer equations. A ( k – ε ) turbulence modeling is used to obtain the eddy viscosity. The finite volume method is introduced to carryout all computational solution with staggered grid arrangement. Due to complex physical domain the original coordinate system is transferred to non orthogonal coordinate system. The calculation of present work done for rotating two dimensional turbine cascade with different rotating speeds (1500 rad/s, 1800 rad/s, 1900 rad/s), and for different Reynolds number (5000, 10000, 100000), in subsonic flow (M<1). The two dimension fluid flow is described by presenting plots of vector and contour mapping for the velocity; pressure and heat transfer fields as well as Nusselt number variation. The results were verified through a comparison with published duct results, good agreement was obtained. The final results were then compared with published results for turbine blades and good agreement was also obtained, the overall comparison show good agreement

the original coordinate system is transferred to non orthogonal coordinate system.The calculation of present work done for rotating two dimensional turbine cascade with different rotating speeds (1500 rad/s, 1800 rad/s, 1900 rad/s), and for different Reynolds number (5000,10000,100000), in subsonic flow (M<1).The two dimension fluid flow is described by presenting plots of vector and contour mapping for the velocity; pressure and heat transfer fields as well as Nusselt number variation.The results were verified through a comparison with published duct results, good agreement was obtained.The final results were then compared with published results for turbine blades and good agreement was also obtained, the overall comparison show good agreement.

INTRODUCTION
The axial flow turbine has two main elements; the stationary vane called nozzle and a turbine rotor.One of the most important aspects of the turbine property is to choose the suitable blade profile.In the three dimensional flow inlets boundary layer separates and forms a horseshoe (or leading edge) vortex, with one leg of the vortex in one aerofoil passage and the other leg in the adjacent passage.Thus in a cascade flow, the part of the secondary flow that is called the passage vortex.
The main objective of present work is to investigate the fluid flow and heat transfer between two rotating turbine blades.This w ill be done by solving the governing continuity, momentum, and energy equations together with the (k -ε) turbulence model, numerically by the finite volume method.An orthogonal curvilinear coordinate system that is rotating with the blade will be used.The development of both hydrodynamic and thermal boundary layers over the blade surface will also be considered.Many investigations have provide data for the flow between turbine blades.However, rather literature is available for flow analysis.Koya and Kotake studied numerically fully developed flow through aturbine stage.Dorfman applied Naveier Stocke equation for gas turbine, to gather with heat conduction.Gogzeh developed finite volume method and code for solving elliptic three dimensional fluid flows.Thomkens study the quazi three dimensional finite difference boundary layer analysis for rotating blade row

MODELING
The blade profile is analyzed in frame work of an orthogonal curvilinear coordinate system rotating with blade.
The geometry under consideration is an impulse turbine blade as represented in figure (1), although it represents a three dimensional flow problem, a suitable simplification by assuming the flow throw a cascade construction will reduce it to a two dimensional problem.The blade dimensions are:-Axial chord, b x = 11.08 in (0.2813m) Chord / axial chord = 1.2242Pitch / axial chord = 0.9555 Aspect ratio (span / axial chord) = 0.9888 In this problem the mean flow is considered to be steady, and the following assumptions were used 1.The corioles acceleration creates a pressure gradient normal to the blade surface u) even though it's larger than in usual boundary layer calculation.2. Assumption of zero pressure gradients across the boundary layer is adopted.
3. It will know assumed that the gradients of all flow property in the x 2 direction are zero 0 x 2    This reduce the x 2 momentum equation with the other equations.4. Incompressible subsonic fluid flow. 5. Pure impulse turbine blade is adopted with zero degree of reaction.6.The boundary layer assume to be turbulent from the leading edge of the blade According to the above assumptions and the flow forces the governing equations will be; (Tompkins, 1982); (1) X 1 Momentum Eq. : (2) X 2 Momentum Eq. : (3) X 3 Momentum Eq.: For the steady state there are four types of boundaries in the physical flow domain, inlet, outlet, solid surfaces and periodic boundary.At the inlet of the blade duct the velocity components and turbulent kinetic energy are specified.Turbulence quantities, such as (k) and (ε) are normally not known, but they must be estimated.Usually (k) is set to :-k=(2/3)(T u .Ux) 2 , ( 6) Where (T u ) is the turbulence intensity and its value (0.01 < T u < 0.1), Davidson (2003).
The dissipation is set to: - Where I c is a characteristic length and is taken as (0.015S), where S is the cascade Pitch.Pressure is assumed to be unchanging in the flow direction at the inlet At the exit plane the values of the dependent variables are unknown.Therefore the outlet boundary should be placed far down from the region of interest

NUMERICAL SOLUTION
In the present study we will work on the sequence of two-dimension grids in successive cross section planes.The equations are formulated in twodimension rectangular compensation domain ) .(   and are solved numerically over uniform, rectangular grid in that domain.The method of Thompson Thames and Mastin (TTM) is utilized.The method employs the following in-homogeneous Laplace eq. as the generation system and its solving results represented in figures (2), ( 3) and ( 4) The table (1) clarifies equation ( 9) in general curvilinear coordinate The Nusselt Number For forced convection of a single-phase fluid where h is called the heat transfer coefficient, with units of W/m 2 , heat flux would be entirely due to fluid conduction through the layer: We define the Nusselt number as the ratio of these two:

Sequence of solution steps
The overall solution procedures can be listed as follows: 1) An initial guess is given for all variables in the field of interest.
2) The proper boundary conditions are specified for all dependent variables.
3) The discretised momentum equation is solved to obtain the covariant velocity components.4) Then the pressure correction equation is solved to obtain pressure correction field.5) The pressure correction is then used to correct pressure fields using equation 6) Then the velocity and density fields are corrected using equation 7) The other dependent variables such as turbulence and energy equations are solved.8) The density is calculated using the new temperature and pressure fields.The whole procedure is repeated from step three until a convergent solution is obtained

RESULTS and Discussion
In the beginning of running the computer program we solve the governing equation for simple rectangular duct as represented in figure ( 5) and ( 6) , the turbulent modeling was also adopted for the duct flow and we can see the turbulent kinetic energy in duct flow in Fig ( 7) and ( 8) The velocity contours of the present work were made for three different values of Reynolds number.
The firs Reynaldos number was 5000 and its (U) velocity contours is presented in fig ( 9), in this figure we can see the uniform velocity at the entrance of the domain, this uniform flow is because the flow coming from uniform source this source is the stationary blade which work as a nozzle to increase the velocity and give the flow more stability.The flow velocity increases uniformly inside the domain and reaches a value over the entrance velocity in the region at the mid chord of the blade and also we see the same in Figures (10) and ( 11).The Vvelocity contour for the three Reynolds number are plotted in figures ( 12) to ( 14), which indicates a uniform velocity at the entrance and it decreases towards the exit The second important variable in the present work is the pressure distribution in the space between the rotating blades.Figures (18), to (20) show the predicted static pressure distribution at different plane for Reynolds number equal (5000, 10000, 100000) respectively, we can see higher pressure distribution near the pressure surface than that close to suction surface this is due to the positive inclination (angle of attack) of the blades The figures ( 21) to ( 23) show the dimensionless turbulent kinetic energy for Reynolds number (5000, 10000, 100000) respectively we observe the minimum value of turbulent kinetic energy near wall and increases toward the center.The turbulent kinetic energy be minimum near wall where the flow velocity is minimum.
Depending on the velocity value appear during solving the governing equation we can get the velocity profile.These profiles shown in Figures ( 25) to ( 28) and for Reynolds Number (100000 and 10000 ) for different locations in the core line for the blades.We fix the points where the velocity reach the max flow velocity we get the boundary layer form as show the Fig ( 29) After predicting the boundary layer profile we fix the point of velocity equal to 0.99 maximum velocity at the suction surface these points provide us with the boundary layer, this process done for 1500 rad/s rotating speed.This procedure repeated for another rotating speed (1800 rad/s, 1900 rad/s) to see how dose the rotating speed affects the boundary layer behavior, the new boundary layer form presented in fig( 30).we see in this figure that increases in rotating speed have bad effect on the boundary layer form specially at (x/c > 0.9) The figures (31 to 34 ) show the temperature distribution developing.For the case of constant upper blade temperature and thermal insulated lower blade for Reynolds (Re = 50,000), we observe that the temperature increases in the direction of the flow and the temperature decreases in the vertical direction.In the middle of the geometry we observe the decrease is greater then decreases slightly toward the lower wall.

CONCLUSION REMREKS
The present work solves the fluid flow and heat transfer between rotating impulse turbine blades.The following conclusions are drawn from the obtained results; the flow velocity increases uniformly inside the calculation domain and reaches a value over the uniform entrance velocity in the region at the mid chord of blade.the velocity vectors are more uniform at the middle section than at the entrance and they become unstable at the trailing edge section.the boundary layers are more uniform at the pressure side than at the section side.
High pressure is observed near the pressure side than thus close to suction side, and denser pressure contours near the leading edge of the pressure side is also observed.A minimum value of turbulent kinetic energy is observed near the wall and it increases towards the center.Increase in rotational speed causes a deformation and increase in boundary layer thickness.A normal temperature distribution is obtained, and the large velocities on the suction side cause streamwise vortex that continuously heat transfer.The Nusselt number is maximum at the entrance section and decreases with the flow direction until it reach approximately constant value.An acceptable agreement was obtained with previous published numerical and experimental results, is the rothalpy H = stagnation enthalpy  = blade row rotation speed R = perpendicular distance from the axis of rotation ±2ρω(υ,ν) = Represent the corioles force component.ρω 2 R = Represent the centrifugal force component With the velocity component normal to the surface neglected.It is convenient at this point to express the Reynolds stress and turbulent energy transport.