Menter/Langtry transition model

This subchapter gives a brief introduction to understand the working principle of the transition model developed by Menter and Langtry and to interpret the numerical results obtained (chapter 4.4.4). A comprehensive study and analysis of this transition model has

been performed in the diploma thesis of Akih Kumgeh.1 The transition model of Langtry and Menter55,62 simulates the transition process described in chapter 2.2.3, but does not attempt to represent the physical nature of the process. The transition model solves two additional transport equations and is performed in three steps (Fig. 3-19). In the first step after an iteration step of the Reynolds averaged Navier Stokes (RANS) equations the critical Reynolds number ReC is determined in each cell with the local flow conditions whereby

this Reynolds number defines the transition onset. In the second step the intermittency controlling the transition length is computed which describes the part of the boundary layer being turbulent thus an intermittency of 0.8 states that the boundary layer is to 80% turbu- lent. At transition onset the intermittency is zero and increases during the transition process to the turbulent value of 1. Outside the boundary layer the intermittency is defined as 1 to consider flow history effects. One example for these flow history effects is the cascade of two turbine blades in which the transition process of the downstream blade is affected by the wake of the upstream blade. In the third step, the evaluated intermittency impacts the SST turbulence model to blend from a laminar to a turbulent solution so that upstream of transi- tion onset the simulation with transitional and laminar boundary layer agree.

0 RANS

2 Intermittency

= f (Conv., Diff., Prod., Destr.) 1 Critical Reynolds ReC

a) Ret = f (RANS)

b) Ret* equation:

Re t*= f (Conv., Diff., Prod.)

3 Turbulence model

Fig. 3-19: Coupling of the transition model with the RANS solver and the turbulence model

As mentioned above, the first step evaluates the critical Reynolds number ReC to deter-

mine transition onset. Therefore, the transition onset Reynolds number Ret and the trans-

ported transition Reynolds number Ret* have to be computed. The transition onset Reynolds

number Ret is based on an empirical correlation taking into account the local flow condi-

tions provided by the RANS solver, the local turbulence level and other local flow condi- tions like e.g. pressure gradient and local flow acceleration. In order to account for flow history effects on transition onset, the transported transition onset Reynolds number Ret* is

evaluated with a transport equation which consists of a convection, a diffusion and a produc- tion term. The convection term accounts for the flow history effects so that the “flow his- tory” is transported along the streamlines outside the boundary layer and the diffusion term spreads this information into the boundary layer which is realized by blending function. The production term is employed to match the “convected” or transported transition onset Rey- nolds number Ret* to the empirical one so that the “flow history” decays with distance to its

origin. The transported transition Reynolds number Ret* finally obtained is employed to

determine the critical Reynolds number ReC which is used in the transition onset criterion

being a function of the local flow conditions, the flow history and an empirical correlation. During the second step a second transport equation for the intermittency is solved which consists of a convection, a diffusion, a production and a destruction term. The convection and the diffusion terms transport the scalar intermittency information which is altered by the production and the destruction terms. The latter decreases e.g. the free stream’s intermit- tency from one to zero at the leading edge of a flat plate to represent the initial laminar boundary layer. The destruction term also allows the relaminarization of a turbulent bound- ary layer e.g. due to strong negative pressure gradients occurring at the intake shoulder (chapter 2.2.3). On the other hand the production term increases the intermittency, if the transition onset criterion is met until the turbulent boundary layer state is reached. Finally, the intermittency is further modified for separated flows so that this influence is also consid- ered in the model.

The transition onset criterion is a key feature of the transition model as it employs only local variables thus allowing parallel computing which is not possible with other transition prediction methods like e.g. the eNmethod.1,81 The transition onset criterion states that the critical Reynolds number ReC has to be equal to the Reynolds number Re determined with

the local boundary layer momentum thickness as reference length. The determination of the momentum thickness would require non-local variables as the momentum thickness has to be evaluated by an integration through several cells. Therefore, for the derivation of the transition model it has been assumed that the momentum thickness Reynolds number Re

could be determined with the maximum vorticity Reynolds number Re,max which employs

only the local variables density, viscosity, the wall distance y and the velocity gradient u/y (eq. (3.13)). The validity of this assumption has been shown for the Blasius profile of a laminar flat plate boundary layer and is presented for hypersonic flow conditions in Fig. 3-20 which displays the maximum vorticity Reynolds number for a hypersonic flat plat flow at Mach 7.5. The good correspondence underlines the assumption holds true also for the hypersonic case.

y u y w w ˜ 2 max , ,Re Re 193 . 2 1 Re PU Q Q T (3.13)

Due to the fact that this transition simulation requires two additional variables to be solved, the total number of equations per cell increases to nine for a simulation with transi- tional boundary layer behaviour. Compared to the five equations required for laminar boundary layer behaviour, the required computational resources are increased leading to the use of the mesh splitting technique.

x [m] R e y n ol ds n u m b e r [- ] 0 0.1 0.2 0.3 0.4 0 500 1000 1500 2000 2500 Re_T Re_Q,max Re_Q,max/2.193

Fig. 3-20: Reynolds number based on the momentum thickness and the maximum vorticity Reynolds number along a hypersonic flat plate flow, Ma = 7.5, Re = 1·107 1/m