A Dual-Grid Hybrid RANS/LES Model for Under-Resolved Near-Wall Regions and its Application to Heated and Separating Flows
Abstract
A hybrid RANS/LES model for high Reynolds number wall-bounded flows is presented, in which individual Reynolds-Averaged Navier-Stokes (RANS) and Large Eddy Simulations (LES) are computed in parallel on two fully overlapping grids. The instantaneous, fluctuating subgrid-scale stresses are blended with a statistical eddy viscosity model in regions where the LES grid is too coarse. In the present case, the hybrid model acts as a near-wall correction to the LES, while it retains the fluctuating nature of the flow field. The dual computation enables the LES to be run on isotropic grids with very low wall-normal and wall-parallel resolution, while the auxiliary RANS simulation is conducted on a wall-refined high-aspect ratio grid. Running distinct, progressively corrected simulations allows a clearer separation of the mean and instantaneous flow fields, compliant with the fundamentally dissimilar nature of RANS and LES. Even with the wall-nearest grid point lying far in the logarithmic layer, velocity and temperature predictions of a heated plane channel flow are corrected. For a periodic hill flow, the dual-grid system improves the boundary layer separation and velocity field prediction both for a constant-spaced and a wall-refined LES grid.
Keywords
Hybrid RANS/LES Turbulence modelling Wall-bounded flows Heat transfer1 Introduction
With the available computational resources of today, turbulence-resolving LES is becoming more attractive for application in industrial purposes. In contrast to statistical RANS schemes, LES enables the prediction of the instantaneous flow field, including pressure and temperature fluctuations, which are determinants of fatigue of critical components in thermal-hydraulics systems. The conduct of LES is however constrained by its high computational cost, which is related to the grid point number N_{xyz} needed to sufficiently resolve the turbulence scale range. For a wall-bounded flow, it is estimated to be exponentially correlated to the flow Reynolds number as N_{xyz} ∝ Re^{13/7} [1], with the majority of the grid points located in the near-wall region in order to resolve anisotropic turbulence therein. In specific, it is the wall-nearest 10% of the boundary layer, where the required grid point number scales with the Reynolds number with high exponent. Piomelli [2] estimates that at moderate Reynolds numbers (Re ≈ 10^{4}), around half of the total grid points have to lie in the inner part of the boundary layer.
To overcome the grid point problem for high Reynolds number flows, the effects of near-wall turbulence can be modelled, what is known as “wall-modelled LES” (WMLES). This set of models includes numerous wall function formulations, which set the wall shear stress in relation to the velocity in the near-wall region. However, justification of wall functions is low for non-equilibrium boundary layers, where the flow does not adhere to the law of the wall. WMLES based on velocity-profile-to-shear-stress correlations cannot be used to predict the wall heat transfer where the Reynolds friction-factor/Nusselt analogy breaks down, e.g. under a wall-impinging jet or at the reattachment point of a flow separation bubble.
Hybrid RANS/LES schemes [3] have been introduced as an alternative to wall functions, where the simulation locally switches from a turbulence-resolving mode to a statistical mode, taking advantage of the weak Reynolds number correlation of the RANS grid resolution requirements. In the context of hybrid RANS/LES, the term “RANS” generally refers to an “unsteady RANS” (URANS) scheme. One of the major hybrid schemes is the Detached Eddy Simulation (DES) family [4]. With multiple development stages, the more recent Improved Delayed DES (IDDES) model [5] is capable of predicting semi-confined flows with high accuracy on grids with low wall-parallel grid resolution, as demonstrated for example for periodic hill and backward-facing step flows [6, 7].
In fact, a cause for these lies in the non-compatibility of averaged and instantaneous variables, which may lead to an excessive damping of turbulent motions in vicinity of the RANS/LES interface. Uribe et al. [16] proposed a hybrid scheme, in which the mean SGS stresses are blended with Reynolds stresses predicted with a RANS model. With the term “Two Velocity Fields” model, they emphasise the different characteristics of instantaneous, fluctuating flow field, and its time-average. Coupling of the LES and RANS model is only undertaken on the mean field level, and the model clearly distinguishes between both flow fields. Results show that turbulence is maintained throughout the near-wall region and is able to transition across the seamless interface.
The differentiating perspective on the fluctuating and averaged flow fields is also found in non-blending approaches: Benarafa et al. [17] use an explicit forcing term in the momentum equation based on the deviation of the statistical RANS velocity from a moving averaged LES velocity field, while Xiao and Jenny [18] extend this concept also to transport equations for the turbulent kinetic energy and eddy dissipation rate. The RANS simulation is considered as the “driving” simulation near walls, while the opposite is true in the main flow region. Similarly, near-wall correction models for the LES Reynolds stresses with modelled RANS Reynolds stresses have been proposed, for example based on forcing terms [19] or changing the constant C_{S} of the dynamic Smagorinsky model [20]. The success of these models, without the use of artificial turbulence at the interface, but by simply treating mean and fluctuating velocity fields separately, strongly stands out.
Passive scalar transport has previously been demonstrated for various hybrid models in different applications, including pin matrix [21], ribbed channel [22] and impinging jet flows [23]. Rolfo, Uribe and Billard [24] presented an extension of the blending concept to the SGS turbulent heat flux as an alternative to the Simple Gradient Diffusion Hypothesis and demonstrated superior prediction of the Two Velocity Fields model to conventional LES for passive scalar transport on coarse grids.
Encouraged by the success of the Two Velocity Fields model, featuring the simple decomposition of SGS stresses, the seamless blending with the RANS eddy viscosity, and the hybrid scalar flux model for heat transfer, we here continue the development of this model. Apart from “Two Layer Models” (TLM) [25], where a supplemental, underlying RANS grid is used to bridge the gap between first LES grid point and the wall, wall-normal grid point reduction in combination with hybrid models is rarely found. A recent and substantially different approach to coarse-grid LES is the “Dual Mesh” model [18, 26], in which LES and RANS simulations are carried out in parallel on two independent grids, each optimised for their respective simulation scheme. A major difference between Xiao and Jenny’s work and TLM is the fully overlapping LES and RANS grid, which makes an a priori sub-domain specification obsolete, and avoids mismatched secondary log-layers at the interface. The present model follows their Dual Mesh approach, but combines it with the Reynolds stress blending method, where the degree of turbulence resolution by the computational grid governs the RANS contribution to the LES. On the other hand, a velocity blending approach feeds back from the LES to the RANS simulation wherever the grid is sufficiently refined. This stands in contrast to Xiao and Jenny’s model, where the LES is by default affected by the RANS at the wall-nearest 20% of the domain.
The remainder of this paper is organised as follows: The present hybrid model is described in Section 2, with the numerical setup given in Section 3. Results for a heated channel flow and periodic hill flow are compared with coarse LES and reference data in Section 4. Conclusions are drawn in Section 5.
2 Model Description
2.1 Blended SGS stress model
Methodical differences exist between the forcing approach by Xiao and Jenny [18, 19] and the present SGS stress blending method. In the original work [18], the instantaneous and mean velocity field are treated with dedicated corrective terms in the transport equations \(Q_{i}^{\text {LES}}=f(U_{i}-\langle \overline {U}_{i}\rangle ,\langle R_{ij}\rangle ^{\text {LES}}-R_{ij}^{\text {RANS}})\). The mean velocity fields of LES and RANS simulations are forced to converge, while the fluctuations are scaled with the difference of the LES and RANS Reynolds stresses \(\langle R_{ij}\rangle ^{\text {LES}}-R_{ij}^{\text {RANS}}\). In an alternative approach [19], the latter is taken as an explicit corrective term to match RANS and averaged LES velocity fields. Main advantage of the forcing approach is the preservation of fluctuation down to the wall.
In the present model, the LES is locally corrected by a RANS eddy viscosity model via the hybrid SGS stress formulation. The total LES stresses gradually transition from \(R_{ij}^{\text {LES}}=R_{ij}^{\text {res}}-2\nu _{t}^{\text {LES}}(\overline {S}_{ij}-\langle \overline {S}_{ij}\rangle )\) for f_{b} = 1 to \(R_{ij}^{\text {LES}}=-2\nu _{t}^{\text {RANS}}\langle \overline {S}_{ij}\rangle \) where f_{b} = 0, in other words where the grid is found to be too coarse to resolve any fluctuation. In practice, the latter case may occur very close to the wall or when the grid is massively coarsened.
2.2 Heat transfer
2.3 Blending function
2.4 LES Eddy viscosity model
2.5 RANS Eddy viscosity model
The original Two Velocity Fields model uses an elliptic relaxation k-ε-φ-f model [32] to model the RANS eddy viscosity \(\nu _{t}^{\text {RANS}}\), which is a robust version of the original 〈v^{2}〉-f model by Durbin [33]. Consideration of the wall-normal Reynolds stress component 〈v^{2}〉 more accurately accounts for the wall blocking effect without using damping functions. f is related to the redistribution of turbulent kinetic energy from the wall-normal to the wall-parallel components and is a solution of an elliptic equation. In the present work, \(\nu _{t}^{\text {RANS}}\) is taken from the BL-〈v^{2}〉/k model by Billard and Laurence [28], which employs the elliptic blending method [34] to further improve the model quality in respect to the Reynolds number and robustness by blending near-wall and homogeneous models of turbulent quantities. This turbulence model is a low-Reynolds model and thus does not require a wall function, but the first grid point layer off the wall to be at y^{+} ≈ 1. The following summarises the model used in the present study. All turbulence quantities of the BL-〈v^{2}〉/k model, in particular the transport equations for k, ε and φ, are computed with the blended RANS velocity field 〈U_{i}〉, which is introduced in Section 2.7.
Model constants of the elliptic blending BL-〈v^{2}〉/k model
C _{ ε1} | C _{ ε2} | C _{ ε3} | C _{ ε4} | σ _{ k} | σ _{ ε} | C _{ μ} | C _{ T} | C _{ L} | C _{ η} | C _{1} | C _{2} | σ _{ φ} |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1.44 | 1.83 | 2.3 | 0.4 | 1 | 1.5 | 0.22 | 4 | 0.164 | 75 | 1.7 | 0.9 | 1 |
2.6 Time-averaging
2.7 Dual-grid simulation
In the present work, the Two Velocity Fields model is extended for the use on grids with low wall-normal resolution. Without relying on a wall function for the RANS turbulence model, the grid has to be sufficiently resolved in wall-normal direction, i.e. to meet the criterion of the wall-normal distance of the wall-nearest cell centre being \(y^{+}_{1}=y_{1}u_{\tau }/\nu \leq 1\) in order to capture a correct velocity-wall shear stress relationship.
3 Numerical Setup
The open-source finite volume solver Code_Saturne [37] is used to implement the proposed hybrid model. This solver is extensively used on high performance computing systems with highly parallelised processes. The code is chosen as it offers a variety of state-of-the-art turbulence models, and as it can handle meshes with any type of cell and topology. To account for the computation of RANS and LES on two grids, simulations are run in parallel and are coupled via Message Passing Interface (MPI) available in the software, which allows the solver to couple with one or more fluid or solid solvers, including other instances of Code_Saturne. L_{t} and \(\nu _{t}^{\text {RANS}}\) are transferred to the LES, while the averaged LES velocity field \(\langle \overline {U}_{i}\rangle \) is returned to the RANS simulation at each numerical time step. The data is interpolated onto the respective grids before the discretised equations are solved. The velocity-pressure algorithm is solved with a segregated prediction/correction scheme, similar to the SIMPLEC scheme [38]. The Rhie and Chow interpolation is used to avoid oscillations. For the solution of the velocity, turbulence transport and passive scalar, a Jacobi system is solved, while a multigrid algorithm is applied for the pressure Poisson equation. The time step Δt is constant and identical for RANS and LES, and is chosen as such that the maximum Courant number is below unity for all simulations. The transport equations of turbulence variables are discretised in time with a second order Crank-Nicolson scheme for the LES, and first order for the RANS simulation. The physical properties are extrapolated with the second-order Adams-Bashforth scheme.
4 Results
4.1 Heated plane channel flow
The fully developed turbulent plane channel flow is computed at friction Reynolds numbers Re_{τ} = u_{τ}δ/ν ∈{395;1,020;2,022;4,079;10,000}. The domain has a size of L_{x} × L_{y} × L_{z} = 6.4δ × 2δ × 3.2δ and is periodic in streamwise (x) and spanwise (z) directions, with walls in the remaining (y) direction. The flow is maintained with a constant pressure gradient in x-direction to obtain the prescribed friction Reynolds number.
Simulation and LES grid characteristics for the plane channel flow at all Re_{τ} numbers
cell spacing | R e _{ τ} | N _{ x} | N _{ y} | N _{ z} | Δ x ^{+} | \({\varDelta } y^{+}_{\delta }\) | Δ z ^{+} | \(y_{1}^{+}\) | Δt ⋅ u_{τ}/δ |
---|---|---|---|---|---|---|---|---|---|
wall-refined | 395 | 40 | 40 | 32 | 63 | 46 | 40 | 1 | 4 × 10^{− 3} |
wall-refined | 1,020 | 40 | 40 | 32 | 163 | 149 | 102 | 1 | 4 × 10^{− 3} |
wall-refined | 2,022 | 40 | 40 | 32 | 323 | 336 | 202 | 1 | 4 × 10^{− 3} |
wall-refined | 4,079 | 40 | 40 | 32 | 653 | 760 | 409 | 1 | 2 × 10^{− 3} |
wall-refined | 10,000 | 80 | 80 | 64 | 800 | 955 | 500 | 1 | 1 × 10^{− 3} |
isotropic | 395 | 128 | 41 | 64 | 20 | 19 | 20 | 10 | 2 × 10^{− 3} |
isotropic | 1,020 | 128 | 41 | 64 | 51 | 50 | 51 | 25 | 1 × 10^{− 3} |
isotropic | 2,022 | 128 | 41 | 64 | 101 | 98 | 101 | 49 | 1 × 10^{− 3} |
isotropic | 4,079 | 128 | 41 | 64 | 204 | 199 | 204 | 100 | 1 × 10^{− 3} |
isotropic | 10,000 | 128 | 41 | 64 | 500 | 487 | 500 | 244 | 1 × 10^{− 3} |
4.1.1 Wall-refined grid
4.1.2 Isotropic grid
The prediction for the mean temperature profile Θ^{+} (Fig. 2b) by the hybrid model is qualitatively lower than for the velocity, with a log-layer slope parallel to the coarse LES and a little dip above the buffer layer. The auxiliary RANS temperature field is not overlapping with the (hybrid) LES simulation, indicating that the velocity coupling is not analogously transferred to the passive scalar field. The mean decomposed Reynolds shear stress component \(R_{12}^{+}\) is shown in Fig. 2c. In case of the hybrid model, the total stresses R_{12} overlap with the reference above y^{+} ≈ 40. On this grid topology, a sharp peak is observed for both hybrid RANS/LES and the coarse LES, where the low wall-normal grid resolution adversely affects the velocity gradient reconstruction. The hybrid SGS model however redistributes the near-wall peak of the SGS shear stress τ_{12}, correcting the observed shear stress at the first grid point in comparison to the coarse LES. This has been analogously observed for the flows at higher Reynolds numbers. The turbulence intensities \(\left (u^{\prime }_{i}\right )^{+}_{\text {rms}}\) (Fig. 2d) are generally well-predicted for all components on the isotropic grid. While both simulations mostly overlap, the intensities are slightly lower near the wall for the hybrid model. This is an indication of minimal turbulence damping effects due to the RANS contribution.
The turbulent heat flux component \(H_{2}^{+}\) (Fig. 2e) overall agrees with the reference. The near-wall peak seen for the Reynolds shear stress is less excessive. In comparison with the coarse LES, the SGS model contribution near the wall is increased. The temperature variance \(\theta ^{\prime +}_{\text {rms}}\) (Fig. 2f) is only slightly lower than the reference away from the wall, yet, the wall-normal resolution is too coarse to return the near-wall peak . Similar to \(\left (u^{\prime }_{i}\right )^{+}_{\text {rms}}\), the comparison to the coarse LES shows small damping effect of the fluctuation due to the RANS contribution.
4.1.3 Higher Reynolds numbers
Relative skin friction coefficient error for the different cases
Grid | Re_{τ} = 395 | Re_{τ} = 1, 020 | Re_{τ} = 2, 022 | Re_{τ} = 4, 079 | Re_{τ} = 10, 000 |
---|---|---|---|---|---|
isotropic | + 3.7% | − 5.5% | − 13.4% | − 3.4% | − 1.7% |
wall-refined | − 4.3% | − 1.7% | − 8.9% | − 6.7% | + 6.5% |
4.1.4 Instantaneous flow field
4.2 Periodic hill flow
The hybrid model is assessed on a channel flow constricted by periodic hills (geometry defined by Almeida et al. [44]) along the bottom wall at a bulk Reynolds number of Re_{b} = 10,595, based on the hill height h and bulk velocity U_{b} at the hill crest. The dimensions of the domain are L_{x} = 9h, L_{y} = 3.036h and L_{z} = 4.5h in streamwise, wall-normal and spanwise direction, respectively. The flow is driven by a mass source term to maintain the desired bulk Reynolds number. The periodic hill flow has been well investigated by others at various Reynolds numbers by means of experiments, DNS and LES. Primary challenge of this flow regime for numerical simulations is the correct prediction of the pressure-induced flow separation on the downstream side of the hill and the resulting shape and length of the recirculation bubble.
Simulation characteristics for the periodic hill flow
Case | N_{xyz}/10^{6} | N _{ y} | max. \(y^{+}_{1}\) | (x/h)^{sep.} | (x/h)^{att.} | Δt ⋅ U_{b}/h |
---|---|---|---|---|---|---|
Reference [45] | 12.4 | 222 | 1.2 | 0.19 | 4.69 | 0.005 |
IDDES (wall-ref.) [7] | 3.1 | 160 | ≈ 1 | 0.19 | 4.55 | 0.015 |
IDDES (const.) [7] | 1.5 | 78 | ≫ 1 | 0.58 | 2.36 | 0.015 |
Dual Mesh [18] | 0.1 | 37 | 36 | ≈ 0.4 | ≈ 4.3 | 0.028 |
Hybrid (wall-ref.) | 0.8 | 80 | 0.9 | 0.25 | 4.42 | 0.027 |
Hybrid (const.) | 0.2 | 50 | 27 | 0.36 | 3.77 | 0.027 |
LES (const.) | 0.2 | 50 | 27 | 0.48 | 2.91 | 0.027 |
4.3 Computational cost
To estimate the additional computational time of the dual-grid hybrid RANS/LES simulations over the coarse LES on one grid, the relative overhead of the average computational time per time step was compared. Both simulation types have been run under identical conditions with maximum 24 cores (two processors with 12 cores each). For the plane channel flow, the hybrid simulations on the isotropic grid and auxiliary RANS grid have been undertaken on 23 + 1 cores, while the coarse LES have been run on 24 cores. The simulations on the wall-refined grids have been run on 7 + 1 cores (hybrid) and 8 cores (coarse LES). In average, a surplus of computational time of around 55% has been observed. The periodic hill flow has been run on 18 + 6 cores for the hybrid RANS/LES and 24 cores for the coarse LES, for which an additional time of 75% has been measured. It was found that the primary factor impacting computational time is the transfer of variables between both simulations, including the grid-to-grid interpolation. Hence, a reduction of the RANS/LES coupling to every tenth numerical time step has been tested, while the auxiliary RANS continues to run in the background. This resulted in a mere overhead of 15% against the baseline LES on the channel flow, with similar results. Further testing of this more economic modification will be part of future work.
5 Conclusion
A hybrid flow simulation method based on synchronised LES and RANS simulations, running on two distinct grids with a seamless Reynolds stress model blending model, is proposed to extend the possibility of partially resolving turbulent-structures with coarse-grid LES to high Reynolds number industrial cases. The dual-grid system, a prolongation of the underlying hybrid Two Velocity Fields model, is shown coherent with the systematic separation of the instantaneous/fluctuating flow field from the time-averaged flow field, as was practised in early 1970’s coarse LES using separate viscosities for mean flow and SGS dissipation [27]. The straddling and roaming of turbulent motions into and out of the RANS-affected near-wall layer is enabled with marginal detrimental impact on RANS and LES flow fields, as displayed by the mean, rms and spectral results, even at the first cell off the wall. The use of the additional RANS turbulence model in the hybrid SGS model is governed by a blending function, which is self-adjusting to the local resolution of turbulence by the grid resolution without reference to the wall distance. The dynamic coupling cycle including a feedback from partial-averaged LES flow field to the auxiliary RANS made both predicted fields statistically consistent. Channel flow and periodic hill test cases showed the versatility of the model for coarse LES with low resolution in wall-parallel, and also in wall-normal directions for a range of Reynolds numbers. The periodic hill flow run on deliberately very coarse grids of 0.2 and 0.8 million cells compared to the literature, produced separation and friction coefficient predictions with reasonable accuracy for industrial purposes.
Without optimisation, the computational cost to run the full hybrid model on the presented test cases is a factor 2-3 times the standalone coarse LES that yields much poorer predictions while halving its mesh step would cost 8-16 times more. Optimising load balancing across parallel computing of the RANS and LES, the extra wall-clock time is reduced to 1.75 when updating coupling terms every time step. A mere 15% extra time-delay was observed when updating these coupling terms only every ten time steps, as these partially time-averaged variables change more slowly than the resolved LES structures and make too frequent data transfer across all cores wasteful. In terms of total CPU, costs are still roughly doubled, but could be reduced if the RANS used a much larger time step than the LES, e.g. in complex geometry design applications with embedded LES.
While the presented framework is at an early stage of development, we identify additional options for further improvement, including the extension to other turbulence models such as second moment closure models. The hybrid model should be further extended to overcome the RANS/LES temperature field discrepancies, possibly by adopting an analogous temperature field blending, using more elaborate scalar flux and temperature variance models and by a Prandtl number dependency of the blending function. Ultimately, the dual-grid scheme may be developed towards an application in an “Embedded LES” context, where the LES grid is refined in flow regions of high interest, with the auxiliary RANS simulation serving for the larger domain.
Notes
Acknowledgements
The present work is supported by the EPSRC and EDF Energy under project number 1652282. The authors declare that they have no conflict of interest.
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