Conceptual model to quantify uncertainty in steady-RANS dissipation closure for turbulence behind bluff bodies

In turbulent bluff body flows, the presence of vortex shedding, a form of coherent structures (CS), introduces a new characteristic scale that is distinct from the scale of background stochastic turbulence (ST). This double-scale picture essentially invalidates the conventional single-scale modeling for the turbulence energy dissipation in steady Reynolds-averaged-Navier-Stokes (RANS) simulations. In this paper, we present a conceptual model to quantify the uncertainty in the steady-RANS dissipation closure for flows past bluff bodies with vortex shedding. This model is developed in two steps. First, we formulate a double-scale, double-linear-eddy-viscosity (DSDL) framework for the transport of CS and ST energy, with an undetermined energy transfer rate from CS to ST. In this framework, the dissipation of ST energy follows a conventional transport model; the length scale of CS is determined such that the CS energy is intensely produced in free shear layers. Second, we design a functional form for the energy transfer rate based on a qualitative analysis and an analogy with the “return-to-isotropy” process. This form contains two uncertain parameters that control the CS-ST interaction. Subsequently, the DSDL model is tested on simulations of the flows past circular and square cylinders, and of the flow in a pin-fin array. In all cases, the model provides a significant improvement upon both conventional models and models with Reynolds stress shape perturbations. The sizes of the recirculation zones are accurately predicted, and simulations with varying values for the uncertain parameters predict intervals for the peak turbulence kinetic energy that encompass reference data.

Figure 1

Principle of the DSDL model: (a) Sketch of a turbulent energy spectrum with coherent structures; (b) tank-tube system analogy for the spectral energy transfer; (c) the supposition of the CS length scale, which naturally yields intense generation of $k^c$ near the separation line behind a structure; and (d) the assumed function to represent the effect of length scale separation on the CS-ST energy transfer.

Figure 2

Mesh for the case of flow past a circular cylinder. Left: in the entire domain; right: near the cylinder.

Figure 3

Mean streamlines and total fluctuation energy contours around the recirculation region behind the circular cylinder (using the $k\text{−}\omega$ SST model as the baseline). $C_{\mathrm{tr}}=1.5$ for DSDL simulations.

Figure 4

Distributions of mean velocity component $U_x$ along the center line $y=0$ downstream of the circular cylinder. (Left: using the low-Re $k\text{−}\varepsilon$ model as baseline; right: using the $k\text{−}\omega$ SST model as the baseline. Same for Figs. 5, 10, and 11.) $C_{\mathrm{tr}}=1.5$ for DSDL simulations.

Figure 5

Distributions of total fluctuation energy $k$ along the center line $y=0$ downstream of the circular cylinder. $C_{\mathrm{tr}}=1.5$ for DSDL simulations.

Figure 6

Logarithmic contours of scale ratio $l^c/l^s$ (left) and CS energy production rate ${\mathcal{P}}^c$ (right) behind the circular cylinder. Simulation DSDL with $C_{\mathrm{tr}}=1.5$ and $r_{1/2}\to \infty$; using the $k\text{−}\omega$ SST model as the baseline.

Figure 7

Isolines of the metrics $m\left(x_r\right)$ (left) and $m\left(k_m\right)$ (right) with respect to parameters $C_{\mathrm{tr}}$ and $r_{1/2}$ for the case of a circular cylinder (using the $k\text{−}\omega$ SST model as the baseline). The axes of $C_{\mathrm{tr}}$ and $r_{1/2}$, respectively, use a logarithmic scale and a reciprocal scale, under which the isolines $m(·)=1$ are linearly fitted by the dashed lines.

Figure 8

Mesh for the case of flow past a square cylinder. Left: in the entire domain; right: around the cylinder.

Figure 9

Mean streamlines and total fluctuation energy contours around the recirculation region behind the square cylinder (using the low-Re $k\text{−}\varepsilon$ model as the baseline). $C_{\mathrm{tr}}=1.5$ for DSDL simulations.

Figure 10

Distributions of mean velocity component $U_x$ along the center line $y=0$ downstream of the square cylinder. $C_{\mathrm{tr}}=1.5$ for DSDL simulations.

Figure 11

Distributions of total fluctuation energy $k$ along the center line $y=0$ downstream of the square cylinder. $C_{\mathrm{tr}}=1.5$ for DSDL simulations.

Figure 12

Logarithmic contours of the scale ratio $l^c/l^s$ (left) and CS energy production rate ${\mathcal{P}}^c$ (right) behind the square cylinder. Simulation DSDL with $C_{\mathrm{tr}}=1.5$ and $r_{1/2}\to \infty$; using the low-Re $k\text{−}\varepsilon$ model as the baseline.

Figure 13

Isolines of the improvement functions $m\left(x_r\right)$ (left) and $m\left(k_m\right)$ (right) with respect to parameters $C_{\mathrm{tr}}$ and $r_{1/2}$ for the case of a square cylinder (using the low-Re $k\text{−}\varepsilon$ model as the baseline). The axes of $C_{\mathrm{tr}}$ and $r_{1/2}$, respectively, use a logarithmic scale and a reciprocal scale, under which the isolines $m=1$ are linearly fitted by the dashed lines.

Figure 14

Computational domain and mesh for the pin-fin array configuration.

Figure 15

Mean streamlines and total fluctuation energy contours on the center plane $z=0$ behind pin rows 2, 5, and 7 of the pin-fin array. $C_{\mathrm{tr}}=1.5$ for DSDL simulations.

Figure 16

Distributions of mean velocity component $U_x$ along the center line $y=0$ on the center plane $z=0$ of the pin-fin array.

Figure 17

Distributions of total fluctuation energy $k$ along the center line $y=0$ on the center plane $z=0$ of the pin-fin array.

Figure 18

Principle of the Reynolds-stress-shape-perturbation (RSSP) method represented in the barycentric map.