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The combined effect of magnetic field, thermal radiation and local suction on the steady turbulent compressible boundary layer flow with adverse pressure gradient is numerically studied. The magnetic field is constant and applied transversely to the direction of the flow. The fluid is subjected to a localized suction and is considered as a radiative optically thin gray fluid. The Reynolds Averaged Boundary Layer (RABL) equations with appropriate boundary conditions are transformed using the compressible Falkner Skan transformation. The nonlinear and coupled system of partial differential equations (PDEs) is solved using the Keller box method. For the eddy-kinematic viscosity the Baldwin Lomax turbulent model and for the turbulent Prandtl number the extended Kays Crawford model are used. The numerical results show that the flow field can be controlled by the combined effect of the applied magnetic field, thermal radiation, and localized suction, moving the separation point, x
_{s} , downstream towards the plate’s end, and increasing total drag,
D . The combined effect of thermal radiation and magnetic field has a cooling effect on the fluid at the wall vicinity. The combined effect has a greater influence in the case of high free-stream temperature.

The idea of controlling the boundary layer flow of an electrically conducting fluid by electromagnetic forces dates back to the 60 s. Rossow was one of the first who studied the incompressible boundary layer flow over a flat plate in the presence of a uniform magnetic field applied normal to the plate [

Recently, the influence of a magnetic field on the flow field has attracted new attention as a control technique for turbulent boundary layers. The magnetic field delays transition from laminar to turbulent flow and separation of the turbulent boundary layer. Transition delay results in a substantial skin friction reduction, since turbulent skin friction is orders of magnitude larger than the laminar one [

The MHD laminar flow in the presence of radiation has been explored by several researchers. Israel-Cookey et al. studied the influence of viscous dissipation and radiation on the problem of unsteady MHD free-convection incompressible flow past an infinite vertical heated plate in an optically thin environment with time-dependent suction [

Thermal radiation has also significant effects on the flow field, especially at high temperatures with important engineering applications. Free convective laminar flow in the presence of radiation has been studied by Ali et al. [

Although many studies exist on the radiation effects on laminar incompressible flows, the study of MHD, compressible, and turbulent boundary layer flow under the influence of thermal radiation and adverse pressure gradient has received little attention. Anghaie and Chen, present a computational model for convective and radiative heat transfer in high temperature gas cooled and gaseous fuel nuclear reactors. Their model considers the turbulent and compressible flow under the effect of radiation in a large range of temperatures [

The subject of flow separation and stability analysis of compressible trailing edge flows has attracted enormous interest in aerodynamics. Turkyilmazoglu, in a theoretical study, has analyzed the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating-disk [

Prevention of flow separation is a challenging task, applicable to several engineering problems. Many passive and active techniques have been developed for the prevention of flow separation [

The goal of this work is the numerical study of the combined effect of the magnetic field, thermal radiation and localized suction on the compressible turbulent boundary layer flow, over a permeable flat plate, in the presence of an adverse pressure gradient. The magnetic field is considered constant and applied to the whole length of the plate. In this study the localized suction, applied to the region of the separation point, is examined. The boundary layer flow is considered turbulent. The electrical conductivity of the fluid is varying with the temperature. The obtained results show that magnetic field, thermal radiation, and local suction influence the flow field and that the separation point is moved downstream to the end of the plate, rendering the above applications as possible flow control techniques.

We consider the steady two-dimensional compressible turbulent boundary layer flow over a smooth and permeable surface. The fluid is a gray, absorbing- emitting radiation, but a non-scattering, medium. It is considered an electrically and heat conducting perfect gas. The plate is an electrical insulator and a magnetic field of uniform strength is applied transversely to the direction of the flow. The magnetic field is assumed to be fixed with respect to the plate and the magnetic Reynolds number of the flow is small so that the induced magnetic field can be neglected [

Continuity equation

energy equation

In the above equations we have replaced the instantaneous quantities

where the subscript,

and the boundary conditions are,

where

where

where

The problem under consideration is described by the system of Equations (11) and (12), subjected to the boundary conditions (13), where the coefficients entering into the equations are defined by the expressions (14). The coefficients

In this study an algebraic turbulent model, Baldwin Lomax model (B-L), for the calculation of the eddy-viscosity and a mathematical model for the turbulent Prandtl number are employed. The B-L is an algebraic model that treats the turbulent boundary layer as a composite layer consisting of inner and outer regions. For the inner region the Prandtl-Van Driest formulation is used. For the outer region, Baldwin and Lomax introduced a formulation that replaces the Clauser formulation of the Cebeci Smith model, avoiding the necessity for finding the boundary layer edge [

The B-L turbulent model was developed for use in multi-dimensional Navier- Stokes codes and the results are in good agreement with the experimental data [

In this study a modification of the extended Kays and Crawford model is used [

In order to study the combined effect of an applied magnetic field, thermal radiation, and localized suction on the flow field a numerical scheme must be applied. The numerical scheme used to solve the parabolic system of PDEs, Equations (11)-(14), is a version of the Keller box method described in [

The free-stream values for the viscosity

In this study we apply a localized suction/injection velocity to a small slot over the plate near the separation point. To examine the influence of local suction we apply a Gaussian distribution [

In MHD boundary layer problems the parameter

The developed numerical code was examined for grid independence [

The most important parameters for engineering applications are the skin friction coefficient,

where

Mach No. | Indices | No MHD/rad./suc. | Combined effect |
---|---|---|---|

1.5 | 5.3966 | 6.0662 | |

1305.1 | 1450.9 | ||

701.1 | 711.8 | ||

2.0 | 5.5865 | 6.0312 | |

2832.1 | 3119.0 | ||

863.4 | 875.6 | ||

2.5 | 5.7764 | 6.1211 | |

5809.1 | 6387.6 | ||

1076.5 | 1090.3 | ||

3.0 | 5.9613 | 6.2461 | |

11,578.6 | 12,741.9 | ||

1341.3 | 1356.5 |

combined effect substantially influences the skin friction coefficient,

The developed numerical code was validated with other published computa- tional results and with experimentally based correlations, showing a good agreement for the case of radiation. More precisely, a comparison with previous computational studies and with experimentally based correlations for a specific problem setup is performed. In this problem, the wall is considered at a steady temperature of 1600 K and the temperature of the free stream is specified at 2000 K, which is a typical design temperature at the core inlet of a gaseous core reactor system [

It is also important to examine the dimensional quantities of the problem under consideration, such as the dimensional velocity and the dimensional temperature. The study of these two dimensional quantities provide a clear image of the shape of the boundary layer under adverse pressure gradient which is very different from the shape of the boundary layer with no pressure gradient.

the maximum temperature increases by 12.2 K (1.5% increase), whereas the temperature decrease (cooling effect) in the boundary layer, close to the wall, is 124 K (17.5% decrease).

The combined effect of magnetic field, thermal radiation, and local suction on the steady turbulent compressible boundary layer flow with adverse pressure gradient is numerically studied. The magnetic field is constant and applied transversely to the direction of the flow. The fluid is subjected to a localized suction and is considered as a radiative optically thin gray fluid. The RABL equations with appropriate boundary conditions are transformed using the compressible Falkner Skan transformation. The nonlinear and coupled system of PDEs is solved using the Keller box method. For the eddy-kinematic viscosity, the B-L turbulent model is used. For the turbulent Prandtl number, the extended Kays Crawford model is used.

The numerical results show that the combined effect of the magnetic field, thermal radiation, and local suction substantially influences the turbulent boun- dary layer, by shifting the separation point downstream to the end of the plate, and increasing total drag. The magnetic field has a greater influence on the flow field in the case of high free-stream temperature. Additionally, the influence of the magnetic field, thermal radiation, and local suction on the thermal boundary layer is significant. The combination of these boundary layer control techniques has a cooling effect on the fluid at the wall vicinity.

The author thanks the reviewer for the valuable comments and suggestions.

Xenos, M. (2017) Thermal Radiation Effect on the MHD Turbulent Compressible Boundary Layer Flow with Adverse Pressure Gradient, Heat Transfer and Local Suction. Open Journal of Fluid Dynamics, 7, 1-14. http://dx.doi.org/10.4236/ojfd.2017.71001