Reynolds-Averaged, Scale-Adaptive and Large-Eddy Simulations of Premixed Bluff-Body Combustion Using the Eddy Dissipation Concept

Abstract

A lean premixed propane/air bluff-body stabilized flame (Volvo test rig) is calculated using the Scale-Adaptive Simulation turbulence model (SAS) and Large-Eddy simulations (LES) as well as the conventional Reynolds-averaged approach (RAS). RAS and SAS are closed by the standard k-𝜖 and the k-ω Shear Stress Transport (SST) turbulence models, respectively. The conventional Smagorinsky and the k-equation sub-grid scales models are used for the LES closure. Effects of the sub-grid scalar flux modeling using the classical gradient hypothesis and Clark’s tensor diffusivity closures both for the inert and reactive LES flows are discussed. The Eddy Dissipation Concept (EDC) is used for the turbulence-chemistry interaction. It assumes that molecular mixing and the subsequent combustion occur in the ’fine structures’ (smaller dissipative eddies, which are close to the Kolmogorov scales). Assuming the full turbulence energy cascade, the characteristic length and velocity scales of the ’fine structures’ are evaluated using different turbulence models (RAS, SAS and LES). The finite-rate chemical kinetics is taken into account by treating the ’fine structures’ as constant pressure and adiabatic homogeneous reactors, calculated as a system of ordinary-differential equations (ODEs) described by a Perfectly Stirred Reactor (PSR) concept. Several further enhancements to model the PSRs are proposed, including a new Livermore Solver (LSODA) for integrating stiff ODEs and a new correction to calculate the PSR time scales. All models have been implemented as a stand-alone application $\text {edcPisoFoam}$ based on the OpenFOAM technology. Additionally, several RAS calculations were performed using the Turbulence Flame Speed Closure model in Ansys Fluent to assess effects of the heat losses by modeling the conjugate heat transfer between the bluff-body and the reactive flow. Effects of the turbulence Schmidt number on RAS results are discussed as well. Numerical results are compared with available experimental data. Reasonable consistency between experimental data and numerical results provided by RAS, SAS and LES is observed. In general, there is satisfactory agreement between present LES-EDC simulations, numerical results by other authors and measurements without any major modification to the EDC closure constants, which gives a quite reasonable indication on the adequacy and accuracy of the method and its further application for turbulent premixed combustion simulations.

Appendices

Appendix A: General Validation of the LSODA and RADAU5 ODEs Solvers

Here, the validation and verification of the RADAU5 [16] and LSODA [17] integrators are provided. Three well-known non-stiff and stiff test problems were chosen for this purpose. The first test is a simple model of flame propagation, which has been introduced in the Matlab ODE suite [68, 69]. The second benchmark is the Van der Pol equation. The third test is the Robertson problem, which consists of a stiff system of three non-linear ODEs for chemical kinetics proposed by Robertson [70]. For sake of completeness, the validation has been provided for both implemented ODE integrators as well as against the state-of-the-art ODE suite available in Matlab [69].

All test cases have been computed with the same absolute and relative error tolerances equal to $10^{-8}$.

1.1 A.1 One dimensional flame propagation

A mathematical model of one-dimensional flame propagation [68] can be described as

$$ \frac{dy}{dt} = y^{2} - y^{3}, \;\; y(0) = \delta, \;\; \left[ 0 < t < \frac{2}{\delta} \right], $$

(49)

where y represents the radius of the sphere, t is time and terms y ² and y ³ come from the surface area and the volume. After igniting, the sphere grows rapidly until it reaches a critical size. Then, the radius of sphere stays at the same size because the amount of oxygen being consumed while combustion in the interior of the sphere balances the amount available through the surface. In this case the critical parameter is the initial radius δ, which determines the stiffness of the problem. A solution becomes stiff near y(t) = 1, increasing or decreasing rapidly toward that solution for small values of δ.

Figure 24 presents the results of integration of this problem for δ = 0.01 and δ = 0.0001. As can be seen, the deviations between all integrators were negligible. All codes computed the problem for both initial conditions.

Fig. 24

Computed solution of the flame propagation problem obtained by RADAU5, LSODA and ode23td (Matlab) solvers

1.2 A.2 Van der Pol equation

The van der Pol equation is a second order ODE:

$$ \frac{d^{2}y_{1}}{dt^{2}} -\mu \left( 1- {y_{1}^{2}} \right) \frac{dy_{1}}{dt} + y_{1} = 0, $$

(50)

where $\mu >0$ is a scalar parameter.

The system of first-order equations can be obtained by making the substitution $dy_{1} / dt = y_{2}$:

$$ \left[ \begin{array}{c} dy_{1} /dt \\ dy_{2} /dt \end{array} \right] = \begin{array}{ll} y_{2} \\ \mu \left( 1 - {y_{1}^{2}} \right) y_{2} - y_{1} \end{array}, \;\; \left[ \begin{array}{c} y_{1}(0) = 2 \\ y_{2}(0) = 0 \end{array} \right], $$

(51)

The nonstiff system (μ = 1) was computed on the time interval [0 20], while the stiff system (μ = 1000) was calculated on the time interval [0 3000]. Figure 25 displays the computed solutions obtained by RADAU5, LSODA and Matlab. All three solutions collapsed well to each other without any significant deviations.

Fig. 25

Computed solutions of the nonstiff (μ = 1) and stiff (μ = 1000) van der Pol equation by RADAU5, LSODA and ode23tb (Matlab) solvers

1.3 A.3 Robertson problem

This problem deals with a system of ODEs that describes the kinetics of an auto-catalytic reaction [70]. The structure of the reactions is

$$\begin{array}{@{}rcl@{}} &&A \overset{k_{1}}{\rightarrow} B\\ &&B + B \overset{k_{2}}{\rightarrow} B + C\\ &&B + C \overset{k_{3}}{\rightarrow} A + C. \end{array} $$

(52)

Under some idealized assumptions [71], the following mathematical model can be set up as a set of three ODEs

$$ \left[ \begin{array}{c} dy_{1} /dt \\ dy_{2} /dt \\ dy_{3} /dt \end{array} \right] = \begin{array}{lllllll} -k_{1} \cdot y_{1} & & k_{3} \cdot y_{2} \cdot y_{3} \\ k_{1} \cdot y_{1} & -k_{2} \cdot {y^{2}_{2}} & -k_{3} \cdot y_{2} \cdot y_{3} \\ & k_{2} \cdot {y^{2}_{2}} & \end{array}, \;\; \left[ \begin{array}{c} y_{1}(0) = 1 \\ y_{2}(0) = 0 \\ y_{3}(0) = 0 \end{array} \right], $$

(53)

where y ₁, y ₂, y ₃ are the concentrations of species $A,B,C$ respectively. The numerical values of the rate constants were k ₁ = 0.04, $k_{2} = 3 \times 10^{7}$ and $k_{3} = 10^{4}$. The large differences among the reaction rate constants provide the reason for stiffness. Originally the problem was proposed on the time interval $0 < t \leq 40$, but it is convenient to extend the integration of solution on much longer intervals due to that many codes fail if t becomes very large.

Figure 26 presents the solutions obtained by RADAU5, LSODA and Matlab. Overall, the discrepancies between calculated solutions were negligible for y ₁ and y ₃. Small deviations between RADAU5, LSODA and Matlab integrators can be observed for y ₂, which probably could be explained by the fact that Matlab has solved the rewritten system of differential algebraic equations by using the conservation law in order to determine the state of y ₃, meanwhile RADAU5 and LSODA calculated the original system of equations.

Fig. 26

Computed solution of the Robertson problem by RADAU5, LSODA and ode15s (Matlab) solvers

Appendix B: Effects of Sub-grid Scaling Modeling on the Inert flow for the Volvo test rig

Three non-reacting LES runs have been carried out to investigate the effects of the SGS models. For this purpose the k-equation and the Smagorinsky model with two different constants (C _s = 0.1 and $C_{s}= 0.053$, respectively were applied. For a quantitative validation of the present LES simulations, the averages have been computed by sampling over 50 vortex shedding periods (N _{v s} ).

Figure 27 shows the measured and predicted mean stream-wise velocity and its fluctuation as well as the normalized turbulence kinetic energy along the central-line behind the obstacle. In general, all three LES runs matched the experimental data by Sjunnesson et al. [19] reasonably well. The discrepancies between all SGS models were negligible. This finding was supported by the fact that all three runs revealed comparatively the same flow patterns shown in Fig. 28. The differences between the LES and the SASI3 results were small as well. The recirculation zone length was predicted as $<L_{r}>/H = 1.28$ for all LES runs (and the same as for SASI3), which was in a fairly good agreement with experimental data of Sjunnesson et al. [19], $<L_{r}>/H = 1.35$.

Fig. 27

Normalized mean stream-wise velocity (a), its fluctuations (b) and and normalized turbulence kinetic energy (c) in the wake centerline for the Volvo test rig

Fig. 28

Flow structures for the Volvo test rig. Iso-surface of the Q-criterion, Q = 1 × 10⁵

Figure 29 compares one-dimensional frequency spectra extracted from the present solutions at the downstream location x/H = 1.75 on the centerline of the wake. About $6\times 10^{5}$ samples of the cross-flow velocities were collected (or $N_{vp} \approx 50$). For sake of completeness, the spectrum obtained by SASI3 was added to assess the dissipative properties of the SAS and LES results. It can be seen clearly that the spectra obtained by LES collapsed well and had the similar distribution, meanwhile the SAS spectrum became more dissipative after f/f _{v s} = 2.5. The Strouhal numbers were $\text {St} = 0.27$, 0.29 and 0.28 for the LESI1, LESI2 and LESI3 runs, respectively. These values were in reasonable agreement with the experimental data by Sjunnesson et al. [19] and Sanquer et al. [56], who measured St = 0.25 and $\text {St} = 0.26$, and corresponded well with the SAS runs.

Fig. 29

One-dimensional spectra of the transverse velocity in the wake for the Volvo test rig: non reactive LES results

Appendix C: Effects of the Sub-grid Scalar flux Modeling for the Volvo test rig

Here, additional inert and reactive LES cases are considered in conjunction with the SGSF closure based on the classical gradient hypothesis closure and Clark’s tensor diffusivity model.

The LESI1 case was chosen as baseline to investigate the influence of the SGSF modeling. As the first step, the inert LES run (LESI1a) was calculated where the diffusion term in Eq. 14 was replaced in the spirit of Clark’s model [67] as

$$ F_{d} = \frac{{\Delta}^{2}}{12} \frac{\partial}{\partial x_{j}} \left( \frac{\partial\tilde{u_{j}}}{\partial x_{j}} \frac{\partial \tilde{k}}{\partial x_{j}} \right). $$

(54)

As the second step, the sub-grid scalar flux was replaced in the energy transport equation as

$$ \text{\textbf{b}}_{h} = \frac{{\Delta}^{2}}{12} \left( \frac{\partial \tilde{u_{j}}}{\partial x_{j}} \frac{\partial \tilde{h}}{\partial x_{j}} \right), $$

(55)

and the inert LES run was performed including both modifications (LESI1b).

Figure 30 compares three cases. Both LESI1a and LESI1b were calculated using identical setup as LESI1. It is clearly seen that differences between all cases were minor, and no clear advantage could be seen when using the particular SGSF model.

Fig. 30

Normalized mean stream-wise velocity and its fluctuations obtained by LESI1, LESI1a and LESI1b runs in the wake of the Volvo test rig

As the third step, the sub-grid scalar flux was replaced in the species transport equation as

$$ \text{\textbf{b}}_{s} = \frac{{\Delta}^{2}}{12} \left( \frac{\partial \tilde{u_{j}}}{\partial x_{j}} \frac{\partial \tilde{Y_{s}}}{\partial x_{j}} \right). $$

(56)

The LESR1 case was chosen to replicate simulations using both Eqs. 55–56 (LESR1a). Figure 31 displays comparison of the normalized, mean temperature and its fluctuations at three axial stages x/H = 0.95, 3.75 and 9.4. In general, the discrepancies between two cases are small. The most pronounced difference was observed at x/H = 0.95, where the LESR1a case provided the slightly lower peak temperature without impulses in the shear layer regions. The minor differences related to the temperature fluctuations were pronounced as well, however, had the same qualitative trends as the baseline case. The species mole fraction results in Fig. 31 were marginally affected, similar to the line thickness or less.

Fig. 31

Normalized mean stream-wise temperature and its fluctuations obtained by LESR1 and LESR1a runs in the wake of the Volvo test rig