Influence of the tangential velocity on the compressible Kelvin−Helmholtz instability with nonequilibrium effects

doi:10.1007/s11467-022-1200-3

Front. Phys.

2022, Vol. 17

Issue (6) : 63500 https://doi.org/10.1007/s11467-022-1200-3

RESEARCH ARTICLE

Influence of the tangential velocity on the compressible Kelvin−Helmholtz instability with nonequilibrium effects

Yaofeng Li¹, Huilin Lai¹(

), Chuandong Lin²(

), Demei Li¹

¹. College of Mathematics and Statistics, FJKLMAA, Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University, Fuzhou 350117, China
². Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China

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Abstract

Kelvin−Helmholtz (KH) instability is a fundamental fluid instability that widely exists in nature and engineering. To better understand the dynamic process of the KH instability, the influence of the tangential velocity on the compressible KH instability is investigated by using the discrete Boltzmann method based on the nonequilibrium statistical physics. Both hydrodynamic and thermodynamic nonequilibrium (TNE) effects are probed and analyzed. It is found that, on the whole, the global density gradients, the TNE strength and area firstly increase and decrease afterwards. Both the global density gradient and heat flux intensity in the vertical direction are almost constant in the initial stage before a vortex forms. Moreover, with the increase of the tangential velocity, the KH instability evolves faster, hence the global density gradients, the TNE strength and area increase in the initial stage and achieve their peak earlier, and their maxima are higher for a larger tangential velocity. Physically, there are several competitive mechanisms in the evolution of the KH instability. (i) The physical gradients increase and the TNE effects are strengthened as the interface is elongated. The local physical gradients decrease and the local TNE intensity is weakened on account of the dissipation and/or diffusion. (ii) The global heat flux intensity is promoted when the physical gradients increase. As the contact area expands, the heat exchange is enhanced and the global heat flux intensity increases. (iii) The global TNE intensity reduces with the decreasing of physical gradients and increase with the increasing of TNE area. (iv) The nonequilibrium area increases as the fluid interface is elongated and is widened because of the dissipation and/or diffusion.

Keywords Kelvin−Helmholtz instability thermodynamic nonequilibrium effect viscous stress discrete Boltzmann method

Corresponding Author(s): Huilin Lai,Chuandong Lin

About author:

Tongcan Cui and Yizhe Hou contributed equally to this work.

Issue Date: 06 September 2022

Cite this article:

Yaofeng Li,Huilin Lai,Chuandong Lin, et al. Influence of the tangential velocity on the compressible Kelvin−Helmholtz instability with nonequilibrium effects[J]. Front. Phys. , 2022, 17(6): 63500.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-022-1200-3
https://academic.hep.com.cn/fop/EN/Y2022/V17/I6/63500

Fig.1 Sketch of D2V16 discrete velocity model.

Fig.2 The position of the sound wave versus time: (a) under different temperatures; (b) under different specific heat ratios. The symbols represent the simulation results, and the solid lines denote theoretical solutions.

Fig.3 (a) Temperature profiles of the thermal Couette flow. The squares, circles, and triangles represent simulation results with

γ = 7 / 5

9 / 7

, and

11 / 9

, respectively. (b) Horizontal velocity profiles of the thermal Couette flow with

γ = 7 / 5

. The squares, circles, upper triangles, lower triangles and diamonds denote simulation results at time instants

t = 0.0

1.0

4.0

8.0

20.0

, and

60.0

, respectively. The solid lines represent theoretical solutions.

Fig.4 Profiles of (a) the density, (b) pressure, (c) horizontal velocity and (d) temperature in the Sod shock tube. The symbols indicate the DBM results, and the solid lines stand for the Riemann solutions.

Fig.5 The initial configuration of the compressible KH instability.

Fig.6 Density and velocity fields in the case of

u 0 = 0.25

at times

t = 0.0

0.5

1.0

1.5

2.0

, and

6.0

, respectively.

Fig.7 Evolution of the whole energies in the KH process. The lines with squares, circles and upper triangles are for the internal, kinetic and total energies, respectively.

Fig.8 (a) Evolution of the global average density gradient in the

x

direction

| ∇ x ρ ¯ |

under different tangential velocities

u 0

. (b) The relationship among

| ∇ x ρ ¯ | m a x

(defined as the maximum of

| ∇ x ρ ¯ |

t (| ∇ x ρ ¯ | m a x)

(defined as the time corresponding to the peak of

| ∇ x ρ ¯ |

) and

u 0

, where the symbols indicate the DBM results, the blue solid line

| ∇ x ρ ¯ | m a x = − 19.78 exp ? (− 5.94 u 0) + 27.87

, and the red solid line

t (| ∇ x ρ ¯ | m a x) = 5.26 exp ? (− 8.43 u 0) + 1.03

Fig.9 (a) Evolution of the global average density gradient in the

y

direction

| ∇ y ρ ¯ |

under different tangential velocities

u 0

. (b) The relationship among

| ∇ y ρ ¯ | m a x

t (| ∇ y ρ ¯ | m a x)

and

u 0

, where the symbols indicate the DBM results, the blue solid line

| ∇ y ρ ¯ | m a x = − 42.67 exp ? (− 4.34 u 0) + 58.49

, and the red solid line

t (| ∇ y ρ ¯ | m a x) = 5.61 exp ? (− 8.26 u 0) + 1.08

Fig.10 (a) Evolution of the global average density gradient

| ∇ ρ ¯ |

under different tangential velocities

u 0

. (b) The relationship among

| ∇ ρ ¯ | m a x

t (| ∇ ρ ¯ | m a x)

and

u 0

, where the symbols indicate the DBM results, the blue solid line

| ∇ ρ ¯ | m a x = − 48.35 exp

(− 4.55 u 0) + 68.03

, and the red solid line

t (| ∇ ρ ¯ | m a x) = 5.27 exp ? (− 7.94 u 0) + 1.05

Fig.11 (a) The simulation results and fitting functions of

| ∇ ρ ¯ |

with various tangential velocities

u 0

at three different times:

| ∇ ρ ¯ | (t = 0.1) = 15.29 + 0.12 u 0 + 0.56 u 02

| ∇ ρ ¯ | (t = 0.5) = 16.13 − 8.93 u 0 + 25.50 u 02

, and

| ∇ ρ ¯ | (t = 0.8) = (0.23 − 1.12 u 0 + 4.66 u 02 − 3.59 u 03) ×

102

. (b) The relationship between

t (| ∇ ρ ¯ |)

(the time corresponding to

| ∇ ρ ¯ |

) and

u 0

at three different values:

t (| ∇ ρ ¯ | = 20) = 2.60 exp ? (− 7.96 u 0) + 0.51

t (| ∇ ρ ¯ | = 25) = 3.65 exp ? (− 8.94 u 0) + 0.61

, and

t (| ∇ ρ ¯ | = 35) = 6.34 exp ? (− 10.71 u 0) + 0.77

Fig.12 (a) Evolution of the global average viscous stress tensor strength

D ¯ 2

under different tangential velocities

u 0

. (b) The relationship among

D ¯ 2 m a x

t (D ¯ 2 m a x)

and

u 0

, where the symbols indicate the DBM results, the blue solid line

D ¯ 2 m a x = (− 0.04 + 2.45 u 0) × 10 − 3

, and the red solid line

t (D ¯ 2 m a x) = 3.41 exp ? (− 6.87 u 0) + 0.85

Fig.13 The simulation results and fitting functions of the global average viscous stress tensor strength

D ¯ 2

with various tangential velocities

u 0

at three different times:

D ¯ 2 (t = 0.2) =

(0.02 + 1.00 u 0) × 10 − 3

D ¯ 2 (t = 0.5) = (0.01 + 0.93 u 0 + 1.02 u 02) ×

10 − 3

, and

D ¯ 2 (t = 0.8) = (0.09 − 0.42 u 0 + 9.09 u 02 − 8.17 u 03) × 10 − 3

Fig.14 Evolution of the global average heat flux strength: (a) in the

x

direction

D ¯ 3, 1 x

, (b) in the

y

direction

D ¯ 3, 1 y

and (c)

D ¯ 3, 1

, with different tangential velocities

u 0

. (d) The relationship among

D ¯ 3, 1 m a x

t (D ¯ 3, 1 m a x)

and

u 0

, where the symbols indicate the DBM results, the blue solid line

D ¯ 3, 1 m a x = [− 0.82 exp ? (− 5.05 u 0) + 1.10] × 10 − 2

, and the red solid line

t (D ¯ 3, 1 m a x) =

5.30 exp ? (− 8.03 u 0) + 1.05

Fig.15 (a) The simulation results and fitting functions of the global average heat flux strength

D ¯ 3, 1

with various tangential velocities

u 0

at three different times:

D ¯ 3, 1 (t = 0.1) = (26.40 − 0.09 u 0 + 0.44 u 02) × 10 − 4

D ¯ 3, 1 (t = 0.5) = (2.70 − 0.63 u 0 + 1.73 u 02) × 10 − 3

, and

D ¯ 3, 1 (t = 0.8) = (0.04 − 1.59 u 0 + 6.29 u 02 − 4.61 u 03) × 10 − 2

. (b) The relationship between

t (D ¯ 3, 1)

and

u 0

at three different values:

t (D ¯ 3, 1 = 0.0035) = 3.02 exp ? (− 8.78 u 0) + 0.59

t (D ¯ 3, 1 = 0.0045) = 4.33 exp ? (− 9.70 u 0) + 0.70

, and

t (D ¯ 3, 1 = 0.0055) = 5.84 exp ? (− 10.53 u 0) +

0.78

Fig.16 (a) Evolution of the global average TNE strength

D ¯

under different tangential velocities

u 0

. (b) The relationship among

D ¯ m a x

t (D ¯ m a x)

and

u 0

, where the symbols indicate the DBM results, the blue solid line

D ¯ m a x = [− 4.53 exp ? (− 3.06 u 0) +

5.40] × 10 − 2

, and the red solid line

t (D ¯ m a x) = 4.81 exp ? (− 7.30 u 0) + 0.99

Fig.17 The simulation results and fitting functions of the global average TNE strength

D ¯

with various tangential velocities

u 0

at three different times:

D ¯ (t = 0.2) = (0.83 +

1.32 u 0) × 10 − 2

D ¯ (t = 0.5) = (0.83 + 1.12 u 0 + 1.66 u 02) × 10 − 2

, and

D ¯ (t = 0.8) = (0.12 − 0.51 u 0 + 3.07 u 02 − 2.51 u 03) × 10 − 1

Fig.18 Contours of the nonequilibrium strength in the case of

u 0 = 0.25

at times

t = 0.02

0.5

1.0

1.5

2.0

, and

6.0

, respectively.

Fig.19 (a) Evolution of the proportion of the nonequilibrium region

S r

under different tangential velocities

u 0

. (b) The relationship among

S r m a x

t (S r m a x)

and

u 0

, where the symbols indicate the DBM results, the blue solid line

S r m a x = [− 2.76 exp ? (− 6.07 u 0) + 3.66] × 10 − 1

, and the red solid line

t (S r m a x) = 5.18 exp ? (− 8.09 u 0) + 1.08

Fig.20 The simulation results and fitting functions of the proportion of the nonequilibrium region

S r

with various tangential velocities

u 0

at three different times:

S r (t = 0.2) =

[− 0.38 exp ? (− 4.54 u 0) + 1.06] × 10 − 1

S r (t = 0.5) = (0.76 + 1.11 u 0) ×

10 − 1

, and

S r (t = 0.8) = [3.43 exp ? (− 3.17 u 0) + 3.56] × 10 − 2

Fig.21 The proportion of the nonequilibrium region

S r

versus the time

t

, with four different threshold values

θ

in the case of

u 0 = 0.25

. The black line, red line, green line, and blue line correspond to threshold values

θ = 0.04

0.05

0.06

, and

0.07

, respectively.

Fig. B1 Grid convergence test of simulations of the KH instability: the global average TNE intensity

D ¯

versus the time

t

, with five different mesh grids. The lines with squares, circles, diamonds, upper and lower triangles correspond to mesh grids

N x × N y = 200 × 200

400 × 400

600 × 600

800 × 800

, and

1000 × 1000

, respectively.

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