Stability of k-ε model in Kolmogorov flow

doi:10.1007/s10483-026-3337-8

Abstract

Abstract:

The Reynolds-averaged Navier-Stokes (RANS) technique enables critical engineering predictions and is widely adopted. However, since this iterative computation relies on the fixed-point iteration, it may converge to unexpected non-physical phase points in practice. We conduct an analysis on the phase-space characteristics and the fixed-point theory underlying the $k -$ $ε$ turbulence model, and employ the classical Kolmogorov flow as a framework, leveraging its direct numerical simulation (DNS) data to construct a one-dimensional (1D) system under periodic/fixed boundary conditions. The RANS results demonstrate that under periodic boundary conditions, the $k -$ $ε$ model exhibits only a unique trivial fixed point, with asymptotes capturing the phase portraits. The stability of this trivial fixed point is determined by a mathematically derived stability phase diagram, indicating the fact that the $k -$ $ε$ model will never converge to correct values under periodic conditions. In contrast, under fixed boundary conditions, the model can yield a stable non-trivial fixed point. The evolutionary mechanisms and their relationship with boundary condition settings systematically explain the inherent limitations of the $k -$ $ε$ model, i.e., its deficiency in computing the flow field under periodic boundary conditions and sensitivity to boundary-value specifications under fixed boundary conditions. These conclusions are finally validated with the open-source code OpenFOAM.

Key words: k-ε model, Kolmogorov flow, instability, turbulence model

2010 MSC Number:

O533

Jiashuo GUO, Le FANG. Stability of $k -$ $ε$ model in Kolmogorov flow. Applied Mathematics and Mechanics (English Edition), 2026, 47(1): 165-184.

TrendMD

Figures/Tables 28

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1

Simulation parameters and configurations of time integration calculation"

Item	Name	Parameter
Governing equation and external parameter	Form of the equation	$k -$ $ε$ model equation
	External parameter	$R e = 2 000,$ 3 000, $S t = 1, 2$
	Model parameter	Standard $k -$ $ε$ model parameters
Computational domain and grid setting	1D domain	$z ∈ [0, L]$
Computational domain and grid setting	Grid generation	1D equidistant grid, $N = 100$ points
Boundary/initial condition	Boundary condition	Periodic boundary conditions
Boundary/initial condition	Initial condition	Uniform fields: $(k, ε) = (0.2, 1),$ (0.4, 1), (0.6, 1), (0.03, 0.001), (0.045, 0.001), (0.06, 0.001)
Numerical method	Discretization	Spatial: 2nd-order central difference
Numerical method	Solver	Temporal: explicit Euler method and time integration algorithm

Table 1

Fig. 5

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Fig. 7

Fig. 8

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Fig. 13

Table 2

Simulation parameters and configurations of cases under periodic boundary"

Setting	Setting name	Parameter
Governing equation and turbulence model	Governing equation	$k -$ $ε$ model equations
Governing equation and turbulence model	Model selection	$k -$ $ε$ model under various parameter conditions
Computational domain and grid setting	Computational domain	3D computational domain $x, y, z ∈ [0, 2 π]$
Computational domain and grid setting	Grid division	3D full hexahedral grid and number of grids $N = 512 000$
Boundary condition and initial condition	Boundary condition	Periodic boundary conditions
Boundary condition and initial condition	Initial condition	Uniform initial fields $k = 0.3$ and $ε = 0.2$
Numerical method and computational setting	Spatial discretization format	Gradient: least squares cell based
		Turbulent kinetic energy: first-order upwind
		Turbulent dissipation rate: first-order upwind
	Time integration format	Explicit Euler method
	Solution algorithm	Time integration
	Convergence tolerance	$10 − 7$
Parameter and model constant	Model parameter	$σ k = 1.0,$ $σ ε = 1.3,$ $C μ = 0.09,$ $(C 1 ε, C 2 ε) = (1.44, 1.92),$ $(1.1, 2), (2.5, 2.8), (1.3, 1.2)$

Table 2

Fig. 14

Fig. 15

Fig. 16

Fig. 17

Fig. 18

Fig. 19

Table 3

Simulation parameters and configurations of cases under fixed boundary"

Setting	Setting name	Parameter
Governing equation and turbulence model	Governing equation	$k -$ $ε$ model equations
Governing equation and turbulence model	Model selection	$k -$ $ε$ model
Computational domain and grid setting	Computational domain	3D computational domain $x, y, z ∈ [0, π]$
Computational domain and grid setting	Grid division	3D full hexahedral grid and number of grids $N = 512 000$
Boundary condition and initial condition	Boundary condition	Fixed value boundary conditions
Boundary condition and initial condition	Initial condition	Uniform initial fields $k = 0.4,$ 0.32, 0.24 and $ε = 0.8,$ 0.64, 0.48
Numerical method and computational setting	Spatial discretization format	Gradient: least squares cell based
		Turbulent kinetic energy: first-order upwind
		Turbulent dissipation rate: first-order upwind
	Time integration format	Explicit Euler method
	Solution algorithm	Time integration
	Convergence tolerance	$10 − 8$
Parameter and model constant	Model parameter	$σ k = 1.0,$ $σ ε = 1.3,$ $C μ = 0.09,$ $(C 1 ε, C 2 ε) = (1.44, 1.92)$

Table 3

Fig. 20

Fig. 21

Fig. 22

Table 4

Simulation configurations of channel flow"

Setting	Setting name	Parameter
Governing equation and turbulence model	Governing equation	RANS equations
Governing equation and turbulence model	Model selection	$k -$ $ε$ model
Computational domain and grid setting	Computational domain	3D computational domain $[0, 4] × [0, 2] × [0, 2]$
Computational domain and grid setting	Grid division	3D full hexahedral grid and number of grids $N = 144 000$
Boundary condition and initial condition	Spanwise boundary condition	Periodic boundary conditions
	Spanwise boundary condition	Fixed boundary conditions
	Streamwise boundary condition	(DNS flow field data of $k, ε,$ and $u$ )
		Periodic boundary condition
		$k :$ kqRWallFunction
	Wall function	$ε :$ epsilonWallFunction
	Wall function	$ν t :$ nutUWallFunction
	Initial condition	Uniform initial fields $k = 2.59 × 10 − 4,$ $ε = 1.5 × 10 − 5,$ and $u = (0.2191, 0, 0)$ (mean values of DNS flow field)
Numerical method and computational setting	Discretization scheme	Gradient: Gauss linear
	Discretization scheme	Divergence: Gauss linear
	Solution algorithm	Semi-implicit method for pressure-linked equation (SIMPLE)
	Convergence tolerance	$10 − 5$
Parameter and model constant	Model parameter	$σ k = 1.0,$ $σ ε = 1.3,$ $C μ = 0.09,$ $(C 1 ε, C 2 ε) = (1.44, 1.92)$

Table 4

Fig. 23

Fig. 24

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