Synthetic tubulence generators (STG) implemented in TRACE:

Shur et al.

Synthetic turbulence can be applied as part of the inlet boundary conditions:

Riemann

and for the following scale-resolving simulation approaches:

Large eddy simulation (LES)
Detached Eddy Simulation
Implicit LES/DNS with Discontinuous Galerkin Method

To use a STG you have to use the control file command

ChgPanelAtt -FAMILY <familyName> -ACTIVATE_STG

You can use both global values and 2D distributions. The corresponding interfaces are summarized in the following table:

STG input variables and possibilities
STG	Necessary input variables	global		2D distribution
Shur et al.	Turbulent length scale \(L_T\)	GMC		Specification of inlet turbulence state
	Reynolds stress \(\uu ij\)	isotrop	Control file command STG_GLOBAL_ISOTROPIC_REYNOLDS_STRESS
		complete tensor	-

Adaption of given total quantities

The Riemann boundary conditions use a given total pressure. Since the added turbulent kinetic energy means an additional energy contribution, the total pressure used by the boundary condition to calculate the mean flow has to be reduced by a corresponding value.

Isentropic Relation:

\begin{align}\frac{p_0}{p}&=\left(1+\frac{\gamma-1}{2}{Ma}^2\right)^{\frac{\gamma}{\gamma-1}}\\ &=\left(1+\frac{\gamma-1}{2}\left(\frac{\left|\mathbf{u}\right|}{a}\right)^2\right)^{\frac{\gamma}{\gamma-1}}\\ &=\left(1+\frac{\gamma-1}{a^2}\frac{\left|\mathbf{u}\right|^2}{2}\right)^{\frac{\gamma}{\gamma-1}}\end{align}

Moving the exponent to the left side and use the mean absolute velocity leads to:

\begin{align}\left(\frac{p_0}{p}\right)^{\frac{\gamma-1}{\gamma}}&= \left[\left(1+\frac{\gamma-1}{a^2}\frac{\left|\mathbf{u}\right|^2}{2}\right)^{\frac{\gamma}{\gamma-1}}\right]^{\frac{\gamma-1}{\gamma}}\end{align}

\begin{align}\left(\frac{p_0}{p}\right)^{\frac{\gamma-1}{\gamma}}&=1+\frac{\gamma-1}{a^2}\frac{\overline{\left|\mathbf{u}\right|^2}}{2}\end{align}

Using the Reynolds average only for the velocity \(u_i =\overline{u_i} + u_i'\) and the relation ([47] [p. 333]):

\begin{align} \frac{1}{2} \overline{u_i^2} &= \frac{1}{2}\left(\overline{u_1}^2 + \overline{u_2}^2 + \overline{u_3}^2\right) + \frac{1}{2}\left(\overline{{u'_1}^2} + \overline{{u'_2}^2} + \overline{{u'_3}^2}\right)\\ &= \sum\limits_i \left(\frac{1}{2}\overline{u_i}^2 + \frac{1}{2}\overline{{u'_i}^2}\right)\end{align}

it follows:

\begin{align}\left(\frac{p_0}{p}\right)^{\frac{\gamma-1}{\gamma}}&=1+\frac{\gamma-1}{a^2}\sum\limits_i \left(\frac{1}{2}\overline{u_i}^2 + \frac{1}{2}\overline{{u'_i}^2}\right)\end{align}

\begin{align}\left(\frac{p_0}{p}\right)^{\frac{\gamma-1}{\gamma}}&=\underbrace{1+\frac{\gamma-1}{a^2}\sum\limits_i \frac{1}{2}\overline{u_i}^2}_{ \left. \left(\frac{p_0}{p}\right)^{\frac{\gamma-1}{\gamma}} \right|_{\text{lam}}} + \frac{\gamma-1}{a^2}\sum\limits_i \frac{1}{2}\overline{{u'_i}^2}\\ &= \left. \left(\frac{p_0}{p}\right)^{\frac{\gamma-1}{\gamma}} \right|_{\text{lam}} + \frac{\gamma-1}{a^2}\sum\limits_i \frac{1}{2}\overline{{u'_i}^2}\end{align}

A similar equation can be determined for the total temperature \(T_0\). The continuity of energy for steady flows with neglected gravity may be written in the following form ([47] [p. 151]):

\[h+\frac{1}{2}\left|\mathbf{u}\right|^2 = h_0\]

The enthalpy of ideal gas is inserted and the equation follows:

\[c_pT+\frac{1}{2}\left|\mathbf{u}\right|^2 = c_pT_0\]

The total temperature calculated with the mean velocity is written as:

\begin{align}T_0 &= T+\frac{1}{2c_p}\overline{\left|\mathbf{u}\right|^2}\\ T_0 &= \underbrace{T + \frac{1}{2c_p}\sum\limits_i \overline{u_i^2}}_{T_{0,lam}} +\frac{1}{2c_p}\sum\limits_i\overline{{u'_i}^2} \end{align}

Similar to the total pressure, the total temperature is reduced by the turbulent part. To non-dimensionalize this equation, the following steps are necessary:

\begin{align} T_0^* &= T^*+\frac{1}{2c_p}\overline{\left|\mathbf{u^*}\right|^2}\\ \frac{T_0^*}{T_0^*}T_0^* &= \frac{T_0^*}{T_0^*}T^*+\frac{1}{2c_p}\overline{\left|\frac{a_0^*}{a_0^*}\mathbf{u^*}\right|^2} \\ T_0^* T_0 &= T_0^*T+\frac{1}{2c_p}\overline{\left|a_0^*\mathbf{u}\right|^2}\\ T_0 &= T+\frac{{a_0^*}^2}{2c_pT_0^*}\overline{\left|\mathbf{u}\right|^2}\end{align}

Since \(a_0^2 = \gamma R T_0^* \) and \(c_p=\frac{\gamma R}{\gamma-1}\)it follows:

\begin{align}T_0 &= T+\frac{\gamma R T_0^*}{2c_pT_0^*}\overline{\left|\mathbf{u}\right|^2}\\ T_0 &= T+\frac{c_p\left(\gamma-1\right)}{2c_p}\overline{\left|\mathbf{u}\right|^2}\\ T_0 &= T+\frac{1}{2}\left(\gamma-1\right)\overline{\left|\mathbf{u}\right|^2}\end{align}