4.7.3 Specifying Inlet Turbulence Levels

It has been observed that the turbulence intensity specified at an inlet can decay quite rapidly depending on the inlet viscosity ratio ( $\mu_t/\mu$) (and hence turbulence eddy frequency). As a result, the local turbulence intensity downstream of the inlet can be much smaller than the inlet value (see Figure  4.7.1). Typically, the larger the inlet viscosity ratio, the smaller the turbulent decay rate. However, if too large a viscosity ratio is specified (i.e., $>$100), the skin friction can deviate significantly from the laminar value. There is experimental evidence that suggests that this effect occurs physically; however, at this point it is not clear how accurately the transition model reproduces this behavior. For this reason, if possible, it is desirable to have a relatively low (i.e $\approx$1 - 10) inlet viscosity ratio and to estimate the inlet value of turbulence intensity such that at the leading edge of the blade/airfoil, the turbulence intensity has decayed to the desired value. The decay of turbulent kinetic energy can be calculated with the following analytical solution:

For the SST turbulence model in the freestream the constants are:

The time scale can be determined as follows:

where $x$ is the streamwise distance downstream of the inlet and $V$ is the mean convective velocity. The eddy viscosity is defined as:

The decay of turbulent kinetic energy equation can be rewritten in terms of inlet turbulence intensity ( $T_{u_{inlet}}$) and eddy viscosity ratio ( $\mu_t/\mu$) as follows:

Figure 4.7.1: Decay of Turbulence Intensity ( $T_u$) as a Function of Streamwise Distance ( $x$)