Hybrid LES/RANS of Internal Flows: A Case for More Advanced RANS

Abstract

The Hybrid LES/RANS is emerging as the most viable modelling option for CFD of real-scale problems, at least in the aerospace design. Entrusting LES to resolve the intrinsic unsteadiness and three-dimensionality in the flow bulk reduces the modelling empiricism to a relatively small wall-adjacent RANS region, arguably justifying the use of very simple models. We argue, however, that for internal flows in complex passages, and involving heat and mass transfer, the role of the near-wall RANS should not be underestimated. The issue is discussed by two examples of flows in turbomachinery: a pinned internal-cooling passage in a turbine blade and tip leakage and wake in a compressor cascade with stagnant and moving casing. The examples illustrate the need for a topology-free wall-integration RANS model that accounts for versatile effects of multiple bounding walls. A HLR using an elliptic relaxation ($\upsilon ^{2}/k-f$) RANS model coupled with a dynamic LES showed to perform well in the cases considered.

Appendix: The HLR \(\zeta \)-f Model

$$ \frac{{D}k}{{D}t}=\mathcal{D}_k +P_k -\alpha \varepsilon ;\quad \quad \frac{\hbox {D}\varepsilon }{{D}t}=\mathcal{D}_\varepsilon +\frac{C_{\varepsilon 1} P_k -C_{\varepsilon 2} \varepsilon }{\tau } $$

$$ \frac{{D}\zeta }{{D}t}=\mathcal{D}_\zeta +f-\frac{\zeta }{k}P_k ;\quad \quad L^{2}\nabla ^{2}f-f=\frac{1}{\tau }\left( {C_1 +C_2 \frac{P_k}{\varepsilon }} \right) \left( {\zeta -\frac{2}{3}} \right) $$

$$\begin{aligned} \alpha =&\max \left( {1,\;\frac{L_{RANS}}{\Delta }} \right) ;\quad L_{RANS} =k^{3/2} /\varepsilon ;\\ \quad \Delta =&C_\Delta \left( {\Delta V} \right) ^{1/3}; \quad \mathcal{D}_\phi =\frac{\partial }{\partial x_j}\left[ {\left( {\nu +\frac{\nu _t}{\sigma _\phi }} \right) \frac{\partial \phi }{\partial x_j}} \right] ;\\ \quad \; \nu _t^{RANS} =&c_\mu \tau \zeta k;\quad \quad \nu _t^{LES} =(c_s \Delta )^{2}\overline{S} ;\quad \quad \nu _t =\max (\nu _t^{RANS} , \nu _t^{LES})\quad \\ \tau =&\max \left[ {\min \left( {\frac{k}{\varepsilon },\frac{0.6{}}{\zeta c_\mu \sqrt{6S^{2}}}} \right) ,c_\tau \left( {\frac{\nu }{\varepsilon }} \right) ^{1/2}} \right] ;\\ \quad L=&c_L \max \left[ {\min \left( {\frac{k^{3/2}}{\varepsilon },\frac{k^{1/2}}{\zeta c_\mu \sqrt{6S^{2}}}} \right) ,c_\eta \left( {\frac{\nu ^{3}}{\varepsilon }} \right) ^{1/4}} \right] \end{aligned}$$

$c_{\mu }$	C $_{\varepsilon 1}$	$C_{\varepsilon 2}$	$C_{1}$	$C_{2}$	$\sigma _{k}$	$\sigma _{\varepsilon }$	$\sigma _{\zeta }$	$c_{\tau }$	$c_{\eta }$	$c_{L}$	$_{C\Delta }$
0.22	1.4(1$+$0.012/$\zeta )$	1.9	0.4	0.65	1.0	1.3	1.2	6	85	0.36	1.5