Compressible Corrections for Turbulence Models in High Speed Boundary Layer Flow

Abstract

Interest in space exploration and high speed travel has renewed interest in vehicles thatare capable of operating in hypersonic conditions. Given the rising demand, there is also need for capable tools to aid in the design of those vehicles. High fidelity methods like direct numerical simulation (DNS) and large eddy simulation (LES) are able to provide very accurate predictions in most conditions because they have little to no reliance on turbulence models which may not accurately capture the physics driving the turbulent motion. The disadvantage of DNS and LES, especially in wall bounded, high Reynolds number flows, is the strict spatial and temporal resolution requirements that quickly drive the computational cost of even simple flows out of the realm of feasibility. The resolution requirement is not easily avoided because it is a physical and mathematical constraint. The alternative to DNS is Reynolds averaged simulations, where the aggregate effectof turbulent fluctuations on the mean flow is modeled instead of being resolved directly. Modeling the effects of turbulence has the effect of greatly reducing the computational expense of the simulation. This allows for the iteration upon and simulation of a design, enabling faster optimization for expected conditions, an obvious advantage for a vehicle designer. The disadvantage is the loss of accuracy relative to direct simulation methods. However, unlike the grid resolution requirements of a DNS, RANS model accuracy is not necessarily a physical constraint; turbulence models can be improved. The current state of RANS turbulence modeling for compressible flows is in need of improvement. Most models currently employed for compressible flows were simply adapted from models developed originally for incompressible flow; in some cases, the original incompressible model is used without any significant modifications. Density-weighted averaging (Favre averaging) has the advantage of making the compressible RANS equations look similar to the incompressible RANS equations, with the same terms in each set. This apparent equivalence has lead to the use of incompressible models in compressible conditions; however, this ignores the physical differences between incompressible and compressible flow and also glosses over assumptions used in the development of the original models, particularly the concept of a universal, invariant boundary layer upon which the model can be built. In this thesis, a framework for compressible corrections is sought. Compressible velocitytransformations that preserve the invariance of the law of the wall over a wide range of Mach numbers and wall heat transfer rates are presented. The compressible velocity trans- formations are then used as the basis for compressible corrections to existing incompressible RANS turbulence models. An implied eddy viscosity is defined and utilized in an algebraic RANS model. The results are compared to existing empirical corrections. The implied eddy viscosity is also used in the equilibrium wall model as part of wall modeled large eddy simulations, because this wall model shares many qualities with algebraic RANS models. For two equation RANS models, the Menter BSL k ω model [1, 2] is explored. Because the implied eddy viscosity cannot be simply inserted into a two equation model, as in the algebraic model, an alternate method based on the compressible correction proposed by Catris and Aupoix [3] is developed. Two equation models developed for incompressible flow also generally do not accuratelypredict the value of turbulence kinetic energy present in the flow. This is fine in incompressible flow where only the value of the eddy viscosity is important and the value of k does not affect the thermodynamic state of the gas. For compressible flow, however, inaccurate k has a negative effect upon the predicted mean temperature in the boundary layer. A viscous correction proposed by Wilcox [4] is tested in compressible flow. The Wilcox correction is also modified using compressible boundary layer scaling arguments and the improvements are noted.