Modeling Heat Fluxes and Aerodynamic Characteristics with Modified Effective Length Method

Aeronautical and Space-Rocket Engineering


Аuthors

Braga A. V.*, Minyushkin D. N.**

Moscow Institute of Physics and Technology (National Research University), 9, Institutskiy per., Dolgoprudny, Moscow region, 141701, Russia

*e-mail: braga.av@phystech.edu
**e-mail: minyushkin.dn@yandex.ru

Abstract

The article presents further development of the modified effective length method intended for engineering estimation of convective heat transfer and aerodynamic characteristics of bodies in the high-speed flow. The proposed approach is stipulated by the need for computationally efficient methods capable of providing engineering-level estimates of the surface thermal loads and force coefficients at the preliminary design stage, when direct numerical resolution of the boundary layer over the entire body surface remains excessively expensive. In contrast to conventional high-fidelity simulations, the method uses local flow parameters at the outer edge of the boundary layer and supplements them with integral characteristics describing the flow prehistory along streamlines. Such a formulation allows retaining principal physical regularities while significantly reducing computational cost.
The existing scheme is being developed in this work in several important directions. Firstly, the modified effective length method is being generalized from the convective heat fluxes calcultaion to the aerodynamic coefficients computing, including the separate accounting for the pressure drag and skin-friction drag. Secondly, engineering models accounting for the surface roughness and wall blowing-in are incorporated into the heat-transfer and transition calculations. Thirdly, allowance is made for this effect impact on the laminar-to-turbulence transition by the criterion based on the Reynolds number. defined with respect to the momentum thickness. This formulation allows introducing conjoint action of roughness and blow-in in physically ostensive and computationally convenient form.
Numerical realization combines two strategies for computing integral characteristics, which are included in this method. In the yielding area, integration is being performed along the reconstructed streamlines on an auxiliary surface, whereas outside this area the required quantities are being recovered by solving a boundary-value problem over the flow field. This hybrid procedure allows avoiding explicit tracing of streamlines over the full surface and improves the method robustness for the bodies of arbitrary geometry. Extra relationships for estimating the boundary-layer thickness, momentum thickness, heat-transfer coefficients in laminar and turbulent regimes, and wall shear stress through a Reynolds analogy-based closure are being introduced.
The developed approach capabilities are being demonstrated on several representative test cases. For a sphere in high-speed flow, the method yields satisfactory agreement for laminar heat-transfer distributions and acceptable agreement for aerodynamic coefficients with reference data. Parametric calculations demonstrate that increasing surface roughness intensifies heat transfer and shifts the laminar-to-turbulent transition toward the upstream region. Blow-in, in its turn, reduces the heat flux in the laminar part of the boundary layer, while shifting transition upstream under the adopted engineering model. The method was applied as well to the cylinders with rounded leading edges, for which satisfactory agreement with reference heat-transfer distributions was obtained. Separate analysis of cylinders with different length-to-diameter ratios reveals that the skin-friction drag contribution to the total drag coefficient increases monotonically with body elongation and should be accounted for in the elongated configurations. Finally, the algorithm was demonstrated for a meteor body of a complex shape, illustrating its applicability to the three-dimensional configurations characteristic to the atmospheric entry problems.
The obtained results indicate that the modified effective length method may be used as an engineering tool for the joint estimate of both thermal and aerodynamic loads on the bodies of complex shape under the high-speed flow conditions. Its main advantage lies in the combination of moderate computational cost, physical interpretability, and sufficient engineering accuracy for parametric studies and preliminary design applications.

Keywords:

effective length method, laminar-to-turbulent transition, aerodynamic coefficients, high-speed flow-around

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