in this paper, the TVD method is applied to the numerical solution of the flow over axisymmetric steady hypersonic viscous flow using TLNS EQUATIONS over blunt cone. In the TVD schemes, the artificial viscosity is implemented using entropy condition. For hypersonic flow, Yee entropy condition has better stability and convergence rate. This paper presents a new entropy condition for increasing the accuracy and convergence rate of the TVD scheme which does not have the difficulty associated with Yee entropy condition for viscous flow in the hypersonic regime. The numerical solution has been compared, with the Beam & Warming shock fitting approach which shows that this approach has better convergence rate and accuracy.

In this work, turbulent supersonic flows over axisymmetric bodies, including the base, are investigated, using multi-block grid to solve the Navier-Stokes EQUATIONS. Patched method has been used near the interfaces. Our numerical scheme was implicit Beam-Warming centeral differencing, while Baldwin-Lomax turbulence modeling was used to close the Reynolds averaged Navier-Stokes EQUATIONS. In order to take advantage of structured grids, multi-block grid has been used widely in the past for complex geometries. On the other hand, different parts of such flow may require different forms of the governing EQUATIONS. For supersonic flows over missile geometries, the thin layer Naveir-Stokes EQUATIONS (TLNS) are usually used. But, the easiest and the fastest approach would be the use of parabolized Naveir-Stokes EQUATIONS (PNS) everywhere. Note, for regions such as the missile’s blunt nose, near fins, and the base flow, we do not have to use the PNS EQUATIONS. Therefore, this leads to the use of both TLNS and PNS EQUATIONS with multi-block solution approaches. The resultes of this new version of our code (MBPTLNS), which is an extension of our original code (MBTLNS) [1], were compared with both computational and experimental benchmark data and showed close agreements.

In this paper, by using implicit fourth order central difference method and TLNS EQUATIONS, the numerical solution of the steady axisymmetric viscous supersonic flow is implemented over blunt cone with shock-fitting method. Because of using high order terms of Taylor series in discretization of derivatives, this method has high accuracy and low numerical error (dispersion error) compared with low order method. The boundary-closure scheme has an important role in stability of this method. By using a coarse grid in this method, the results of numerical solution are found to be very close to those obtained with a fine grid employing the second order (Beam-Warming) method. Higher accuracy of this method is identified relative to the second order method when the grid is being refined. The convergence of this method can be adjusted to accommodate the computational hardware capabilities.

A turbulent supersonic flow over bodies of revolution, including the base flow, is investigated using multiblock grid to solve the thin layer Navier-Stokes (TLNS) EQUATIONS. Patched method has been used near the interfaces. Our numerical scheme was implicit Beam-Warming central differencing, while Baldwin-Lomax turbulence modeling was used to close the Reynolds averaged Navier-Stokes EQUATIONS. The results of this work have been compared with existing computational and experimental benchmark data and shows close agreements.

In this work, a supersonic turbulent flow over a long axisymmetric body was investigated, both experimentally and computationally. The experimental study consisted of a series of wind tunnel tests for the flow over an ogive-cylinder body at a Mach number of 1.6 and at a Reynolds number of 8´106, at angles of attack between -2 and 6 degrees. It included the surface static pressure and the boundary layer profile measurements. Further, the flow around the model was visualized using a Schlieren technique. All tests were conducted in the transonic wind tunnel of the Qadr Research Center (QRC). Also, the same flow at zero angle of attack was computationally simulated using a multi-block grid (with patched method around the block interfaces) to solve the thin layer Navier-Stokes (TLNS) EQUATIONS. The numerical scheme used was implicit Beam and Warming central differencing, while a Baldwin-Lomax turbulence model was used to close the Reynolds Averaged Navier-Stokes (RANS) EQUATIONS. The static surface pressure results show that the circumferential pressure at different nose sections varies significantly with angle of attack (in contrast to the circumferential pressure signatures along the cylindrical part of the body), while the total pressure measurements in the boundary layer vary significantly both radically and longitudinally. Two belts with various leading edge angles were installed at different locations along the cylindrical portion of the model. The computational results obtained were compared with some experimental ones (found by these authors), showing considerably close agreements.

The steady state flow of viscous fluid in two dimensional and axisymmetric nozzles in all regimes has been numerically studied using compressible TLNS EQUATIONS. Implicit delta form of Roe finite difference is used as the numerical scheme. The numerical simulation of two-dimensional transonic flow through a double throat nozzle is studied as a test problem. In order to show ability of the method flow in a plug nozzle is also studied. Due to the strong interaction of shock, the entropy condition in solution of plug nozzle has been modified. The dependency of the solution upon grid refinement has been also studied and the results are validated using the results of previous investigators.

The three-dimensional supersonic turbulent flows over wrap-around fin missiles have been computed using the Thin Layer Navier-Stokes (TLNS) EQUATIONS to reduce the computational efforts compared to those of the Full Navier-Stokes (FNS) EQUATIONS. In this research, the missile configuration is divided into multi regions to enable fluid flow simulation using Personal Computers (PC). It also makes it possible to use a different number of nodes and distribution of grids in each region to enhance the accuracy. The Thin Layer Navier-Stokes EQUATIONS in the generalized coordinate system were solved using an efficient, implicit, finite-difference factored algorithm of the Beam and Warming. For the turbulent flow field computations, the well-known isotropic two-layer algebraic eddy viscosity Baldwin-Lomax model was used. The computations were performed for supersonic turbulent flows over wrap-around fin configurations for free stream Mach numbers, M_{\infty}=1.9-2.86. Predicted roll moment and longitudinal aerodynamic coefficients were compared with the experimental data at various angles of attack. The computational results are in good agreement with experimental data.