Separation of turbulent boundary layer in the presence of adverse pressure gradient is encountered by the many engineering applications, such as high angle of attack airfoils, diffusers, and turbomachinery. Flow separation can be a significant factor of loss of lift and increase of drag, in other words drop of the performance of the machines.

Computational simulation of turbulent flows has many difficulties to deal with. Besides the complex nature of turbulence, separation makes it harder to deal with these flows. The equations of motion require modeling assumptions in order to determine the effects of turbulence to the mean momentum equations, which is an additional consideration beside the computational efficiency and accuracy of the computational fluid dynamic work. The turbulence models can be the main source of error in the calculations and it is difficult to determine whether the turbulence model or the numerical methods are responsible from the difference between the experiment and numerical solution.

There exist many turbulence models in use, however they supply accurate results in a limited range of flows. The model of Mellor and Yamada (1977) will be used in this study. It is consists of transport equations for Reynolds stresses and length scale.

In this study vorticty stream function approach is choosen. Once the vorticty vector is obtained Poisson’s equation is used to calculate stream function, which will be used to determine the mean velocity components.

The wall layer model, that was proposed by Mellor 1966 to determine the wall shear stress, will be used. This model has significant superiority over classical law of the wall.

Çelenligil's 1981 and Gülbahçe's 1994 calculations needs some improvement, Both Çelenligil and Gülbahçe used relatively coarse, uniform meshes and it is thought to be the source of the significant error. In order to obtain the desired resolution, without producing unnecessary gird points, use of stretched rectangular mesh is decided.

Another method to increase the accuracy of the calculations is changing the numerical scheme. The piecewise polynomial reconstruction with adaptive stencil is designed to use, which is third order accurate. The adaptive stencil is due to preventing oscillations at the presence of large gradients.

Improvements on the Numerical Methods

Basic aim of this study is to increase the accuracy of the calculations to obtain the performance of the turbulence closure model in separated flows. In order to satisfy this objective, some improvements are needed for the studies of Çelenligil and Gülbahçe in an ordered way.

Both the accuracy and the efficiency of the calculations can be increased significantly by employing a stretched mesh coordinate system instead of a uniform mesh. Using stretched mesh, critical areas of the flow can be calculated with high resolution, and less critical regions can be calculated with coarser mesh.

In the study of Gülbahçe various numerical schemes were adapted for convective parts of the equations, such as first order upwind, leapfrog, Arakawa and Leonard methods. In employing last three schemes explicit numerical dissipation (artificial viscosity) have to be added whereas in first order upwind method it is added implicitly. This dissipative terms effects the accuracy of the solution.

In order to increase the accuracy more advanced methods are investigated and essentially Non-Oscillatory method is chosen. The solution of the linear advection equation at 16667-time step is given for both discontinuous and smooth function transport is presented in Slide 7 and Slide 8.

This method creates essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution over each cell. The reconstruction algorithm is derived from a new interpolation technique that, when applied to piecewise smooth data, gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. Different from the other methods this procedure uses an adaptive stencil of grid points.

There exist two ways to handling the second derivatives or diffusive terms, it can be absorbed by the convective part and treat using ENO. The second procedure is using standard high order central differencing schemes to compute second derivatives.It was state that there is little difference between between them so latter one is accepted because of simplicity.

The evolution of time integration is another important subject to deal with. Again there are two approachs one is Lax-Wendroff technique other is method of lines. Since as the order of accuracy increases the first method becomes very very complicated., the latter one is decided to use. In order to increase accuracy and preserve stability TVD Runge-Kutta methods are decided to use. The superiority of TVD Runge-Kutta over classical non-TVD one is given in Slide 9 from [4].

By applying these methods it is desired not to add explicit disspative terms for sake of stability for both spatial and temporal integrations(which were highly used in previous studies).

Since finding the vorticity at time level n+1, the stream function can be determined by using the Poisson equation. The Poisson equation was solved by using the iterative Alternating Direction Implicit method , ADI. This method well suited to the time dependent solutions. The extension to stretched mesh coordinates is staright forward.

CONCLUSION

It is supposed that after applying these changes to the previous desired accuracy would be obtained. In the ongoing study the spatial integration procedure for convective terms can be improved with weighted essentially non-oscillatory (WENO) schemes.

Also some analytical improvements can be made. For instance wall turbulent model can be changed with an improved one proposed by Smith.

REFERENCES

2-Çelenligil MC 1982 "Numerical simulations of Incompressible Planar Turbulent Separated Flows", Ph. D.Thesis, Department of Mechanical and Aerospace Engineering, Princeton University.

5-Gülbahçe R 1994 "Numerical simulations of Two-Dimesional Incompressible Turbulent Flows" MS Thesis, Department of Aeronautýcal Engineering, METU

6-Laney CB 1998 "Computational Gasdynamics" Cambridge Unýversity Press.

8-Smith BR 1987, "Calculation of Separated Non Stationary Two Dimensional Turbulent Flows Around Bluff Bodies Using Reynolds Stress Closure Turbulence Modeling"Ph, D, Thesis, Department of Mechanical and Aerospace Engineering, Princeton University.