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TRACE User Guide
TRACE Version 9.6.1
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TRACE User Guide
TRACE Version 9.6.1
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It is a well-known problem of two-equation Boussinesq models that an unphysical overproduction of turbulent kinetic energy takes place in stagnation regions. Various proposals to remedy this problem have been made. The stagnation point fixes implemented in TRACE are described below.
Based on the idea that in the entire flow region, except for stagnation regions,
\begin{eqnarray} \left|S \right| \approx \left| \Omega \right| \end{eqnarray}
(cf. Kato & Launder [22] ), the turbulence model’s production terms are modified according to
\begin{eqnarray} \tau_{ij} \frac{\partial u_i}{\partial x_j} = \mu_T \left| S \right|^2 \approx \mu_T \left| S \right| \left| \Omega \right| \end{eqnarray}
Based on the Cauchy-Schwarz inequality
\begin{eqnarray} \left( \overline{u_i' u_j'} \right) \left( \overline{u_i' u_j'} \right) \leq \overline{u_i'^2 u_j'^2} \end{eqnarray}
a lower bound for \(\omega\) can be created, so that
\begin{eqnarray} \omega = \max \left( \omega, \frac{1}{2} \sqrt{3} \cdot \sqrt{2 S_{ij} S_{ij}} \right) \end{eqnarray}
This modification is effective at the update of turbulence quantities, rather than being a modification of the model equations themselves.
When Favre-averaging the equations of motion, and entering Favre-averaged quantities into the definition of turbulence quantities, some additional precautions need to be taken [67] .
Phenomena associated with compressibility effects include dilatation dissipation, pressure dilatation and pressure work. The next section describes some details regarding the modeling of these effects.
This addition corrects the prediction of mixing layer growth in compressible flow, which is not properly predicted by standard formulations starting at
\begin{eqnarray} M > \approx 0.5 \end{eqnarray}
This is achieved by modifying the destruction terms’ closure coefficients [65], [66] .
\begin{eqnarray} \beta_{k,c} = \beta_k \left[ 1 + \frac{3}{2} \max \left( M_t^2 - \frac{1}{16}, 0 \right)\right] & \Rightarrow & \varepsilon_{c}= \left[ 1 + \frac{3}{2} \max \left( M_t^2 - \frac{1}{16}, 0 \right)\right] \varepsilon \\ \beta_{\omega,c} = \left[ \beta_\omega - \frac{3}{2} \beta_k \max \left( M_t^2 - \frac{1}{16}, 0 \right)\right] & \Rightarrow & D_{\omega,c}= \left[ 1 - \frac{3}{2} \frac{\beta_k}{\beta_\omega} \max \left( M_t^2 - \frac{1}{16}, 0 \right)\right] D_\omega \end{eqnarray}
Via Favre-averaging, two additional terms for pressure dilatation and pressure work appear in the exact \(k\)-equation.
These are modelled following a scheme by Sarkar [46] by changing the turbulence model equations as follows (only source terms are shown):
\begin{eqnarray} \frac{\partial}{\partial t} \left( \rho k \right) &=& \left(1 + \alpha_2 M_t \right) P_k - \left(1 - \alpha_3 M_t^2 \right) \varepsilon \\ \frac{\partial}{\partial t} \left( \rho \omega \right) &=& \left[ C_{\varepsilon 1} - \left(1 + \alpha_2 M_t \right) \right] \frac{\omega}{k} P_k - \left[ C_{\varepsilon 2} - \left(1 - \alpha_3 M_t^2 \right) \right] \varepsilon \end{eqnarray}
Note that the modification of the \(\omega\)-equation appears when transforming the model modification from a \(k\)- \(\epsilon\) to a \(k\)- \(\omega\) formulation.
When comparing the turbulence model for the decay of homogeneous turbulence in uniform rotation by Bardina et al. [3] to other two-equation transport models, an additional destruction term appears in the scale determining equation. Transforming this term to a \(k\)- \(\omega\) -notation yields (all other terms omitted)
\begin{eqnarray} \frac{\partial}{\partial t} \left( \rho \omega \right) = - c_2 \rho \omega \Omega_{\mathrm{norm}} \end{eqnarray}
The closure coefficient \(c_2\) was calibrated to match experimental data.
Note: This addition is switched off automatically in transition-active regions.
The effect of curved streamlines on the amplification (concave streamlines) or damping (convex streamlines) is modelled with an additional source term on the right-hand side of the \(\omega\)-equation [26] :
\begin{eqnarray} D_{\mathrm{SLC}} = C \kappa_{\mathrm{long}} \rho \omega \sqrt{u_i u_i} \end{eqnarray}
with
\begin{eqnarray} C = f \left( r^* \right) \begin{cases} > 0 \; ; 0 < r^* < 1 \\ = 0 \; ; r^* = 1 \\ < 0 \; ; r^* > 1\end{cases} \end{eqnarray}
and
\begin{eqnarray} r^* = \frac{2 S_{ij} S_{ij}}{2 \Omega_{ij} \Omega_{ij}} \end{eqnarray}
Flow solutions with the two-equation turbulence model show a much to deep and narrow wake down-stream of the blade profile on a grid with sufficient resolution of the velocity gradients. Flow separations at the trailing edge generate self-induced coherent structures dependent of the size of the edge thickness. This phenomenon is known as the Karman vortex street. It is obvious that the series of huge vortices gives rise to more intense mixing in comparison to the vortices of a turbulent boundary layer. A detector was sought and derived from the observation that there is an entrainment of fluid from the meanflow into the wake region near the profile trailing edge. In this region strong total pressure gradients exist. For the inviscid incompressible flow the total pressure gradient can be represented by the vector-product of stream-wise velocity and vorticity.
\begin{eqnarray} \mathbf{u} \times \left( \bigtriangledown \times \mathbf{u} \right)= \bigtriangledown \left( \frac{p_t}{\rho} \right) \end{eqnarray}
If this vector is multiplied by the normal velocity vector relative to a flow target direction, then a scalar appears which could be called normal energy.
\begin{eqnarray} nE= \left( \mathbf{u} \cdot \mathbf{e_n}\right) \cdot \mathbf{u} \times \left( \bigtriangledown\times \mathbf{u} \right) \end{eqnarray}
The evaluation of negative normal energy shows, that in regions of appearance of the Karman vortex street occur negative normal-energy. Therefore a modeling was developed that in regions with negative normal energy is allowed more diffusion of turbulent kinetic energy and less diffusion of specific dissipation rate. Also, the constants for the production- and for the dissipation source term of specific dissipation rate was fitted for the model in a similar way as for the Menter SST model of the outer layer.
The modeling on pure constants provided marginal stability improvements in comparison to the distribution-parameter on the model invariant non-dimensional strain rate \(S_{ij}\) or the parameter for distribution of the local ratio of characteristic macro time scale of turbulence \(t_t\) and the characteristic time scale of mean velocity field \(t_m\)
\begin{eqnarray} S_{ij} \equiv \frac{t_t}{t_m}= \frac{1}{\beta_{k,Wilcox1988} \cdotp \omega} \cdotp \frac{1}{2} \left( \parfrac{U_i}{x_j}+ \parfrac{U_j}{x_i} \right). \end{eqnarray}
The flow target direction or rather the quasi-hardware direction (basically an extension of the camber-line of the blade) has to be provided for each cell in the same way as the wall distance for the Menter SST or the Gamma-ReTheta transition model.
To reflect the limited occurance of vortex streets at low and higher trailing edge diameter Reynolds-Numbers the model is only active at Reynolds numbers higher than 60 and lower than 30000 with a exponential blending function from 10000 to 30000.
The comparisons of wake-shape results of laminar and transitional cascades for the standard Wilcox1988 k-omega-model and for the standard Menter SST model with the model of locally increased diffusion show the improvements. The wake-width and -depth agree more with the measurements.
The quasi-hardware direction assumes a certain structured block topology. It is performed within the fall-back wall distance computation inside of TRACE. If the model is active, additional fields are written to the CGNS-File (non dimensional strain norm, normal energy, quasi-hardware direction).
For the Menter SST \(k\)- \(\omega\) model, as outlined in [30], a Durbin realizability constraint can be used instead of the production bound in the original model. The eddy viscosity is thus defined as
\begin{eqnarray} \mu_T = \rho k T \; \; \; \text{with} \; \; \; T = \min\left[\frac{1}{\max\left(\omega,\frac{S F_2}{a_1}\right)},\frac{0.6}{\sqrt{3} S}\right] \end{eqnarray}
The extension is activated via Turbulence model sub-options.
For the Menter SST \(k\)- \(\omega\) model, in order to remedy the reattachment location being predicted too far downstream, a modification has been suggested by Menter which is presented in [6] . The production term of the \(k\)-equation is altered as follows:
\begin{eqnarray} P_{reattach}= P_k \min \left[4\max\left(0,\frac{\min(S^2,\Omega^2)}{0.09\omega^2}-1.5\right),1.5\right] \end{eqnarray}
The extension, however, has achieved mixed success. It is activated via Turbulence model sub-options.
For external aerodynamics applications, the decay of turbulence from the farfield boundary to the geometry is often excessive and unphysical. Additional source terms can be introduced to the \(k\)- and \(\omega\)-equations of the Menter SST model which sustain an ambient level of turbulence [52] .
\begin{eqnarray} S_k &=& \beta^* \rhobar k_\text{amb} \omega_\text{amb} \\ S_\omega &=& \beta \rhobar \omega_\text{amb}^2 \end{eqnarray}
with the ambient turbulent kinetic energy \(k_\text{amb}\) and dissipation rate \(\omega_\text{amb}\). The model constants are blended as in the original model.
The extension is activated via Turbulence model sub-options.
For the \(k\)- \(\omega\) model, there is the Viscous Blending Limiter, an extension that limits the eddy viscosity and thus the production of turbulent kinetic energy in turbulent flows. The eddy viscosity formulation is changed to:
\begin{eqnarray} \mu_T = \frac{a_1 \rho k}{max(a_1 \omega,\; b_v S)} \end{eqnarray}
with \(a_1 = 0.31\) and \(b_v\) being a detection factor for areas of viscous flow that depends on the model version. \(S\) denotes the traceless strain tensor norm of the flow. For Viscous Blending version 1, \(b_v\) is defined as:
\begin{eqnarray} b_v &=& min \left( max \left( \frac{\omega_{Quot}}{500},\; 0.1 \right),\; 1 \right) \\ \omega_{Quot} &=& \omega / \omega_{corr} \\ \omega_{corr} &=& k^{-0.613} \end{eqnarray}
while in version 2, \(b_v\) is defined as follows:
\begin{eqnarray} b_v &=& min \left( max \left( \frac{2 \: atan( \frac{\omega}{\omega_{corr}} + c_2)}{\pi} - c_3\: f_{VRPM,w}, 0 \right), 1 \right) \\ \omega_{corr} &=& \omega_{ref} \left( \frac{k}{k_{ref}} \right)^{-0.613} c_1 \\ f_{VRPM,w} &=& min \left( max \left( f_{VRPM} \frac{\Omega_{SW}^2}{\Omega^2}, 0 \right), 1 \right) \end{eqnarray}
\(c_1 = 1, c_2 = 0 \text{ and} c_3 = 2\) are model constants, the quantities \(k_{ref}\) and \(\omega_{ref}\) are reference values that are defined as the area average of the respective turbulence model quantities at the blockgroup inlet panel. Hence, these reference values are computed for each row of a multistage computation. The factor \(f_{VRPM,w}\) is based on the turbulent kinetic energy production factor of the VRPM turbulence extension which is multiplied with the weight of the streamwise vorticity \({\Omega_{SW}^2}\: / \:{\Omega^2}\). This extension to the computation of the \(b_v\) factor is necessary to reduce the eddy viscosity limiting in side wall vortices.
Viscous Blending is activated via Turbulence model sub-options.
The Menter SST \(k-\omega\) model [35] has its origin in the Baseline model. The main difference of the Baseline model and the Menter SST model is that the eddy-viscosity is not calculated using the Bradshaw assumption.
\begin{eqnarray} \mu_{T} = \frac{\rho k}{\omega} \end{eqnarray}
In addition, the constant \(\sigma_{k1}\) is set to \(\sigma_{k1}=0.5\).
The Baseline model does not constrain the production of turbulent kinetic energy, which is important, e.g., for surface roughness.
The use of the Hellsten limiter [18] is recommended when accounting for surface roughness with the Menter SST \(k-\omega\) model [35] . It deactivates the Bradshaw assumption in the near-wall region with an additional Parameter $F_3$ in the eddy-viscosity formulation.
\begin{eqnarray} \mu_{T} = \frac{\rho a_1 k}{\mathrm{max}(a_1 \omega, S F_2 F_3)} \end{eqnarray}
The Bradshaw assumption was first introduced in the SST \(k-\omega\) model to avoid excessive shear-stress, which can be a problem of Boussinesq eddy-viscosity models.
1.9.1