Numerical validation of novel scaling laws for air–water flows including compressibility and heat transfer

Figures & data

Table 1. NSLs for the variables of the governing equations for air–water flows including compressibility and heat transfer

	Scaling conditions in terms of ${\alpha }_x$, ${\alpha }_t$, ${\alpha }_{\rho }$, ${\alpha }_T$ and ${\alpha }_n$		Scaling conditions in terms of ${\alpha }_x$, ${\alpha }_{\rho }$, ${\alpha }_g={\alpha }_R=0$ and ${\alpha }_n$		Scaling conditions in terms of ${\alpha }_x$, ${\alpha }_R={\alpha }_T={\alpha }_{M_m}=0$ and ${\alpha }_n=3{\alpha }_x$
Variables	Exponents	Scaling ratios	Exponents	Scaling ratios	Exponents	Scaling ratios
$x_i$ (m)	${\alpha }_x$	${\text{β}}^{{\alpha }_x}$	${\alpha }_x$	${\text{β}}^{{\alpha }_x}$	${\alpha }_x$	${\text{β}}^{{\alpha }_x}$
t (s)	${\alpha }_t$	${\text{β}}^{{\alpha }_t}$	${\alpha }_t=0.5{\alpha }_x$	${\text{β}}^{0.5{\alpha }_x}$	${\alpha }_t={\alpha }_x$	${\text{β}}^{{\alpha }_x}$
ρ (kg ${\mathrm{m}}^{-3}$)	${\alpha }_{\rho }$	${\text{β}}^{{\alpha }_{\rho }}$	${\alpha }_{\rho }$	${\text{β}}^{{\alpha }_{\rho }}$	${\alpha }_{\rho }=0$	${\text{β}}^0=1$
T (K)	${\alpha }_T$	${\text{β}}^{{\alpha }_T}$	${\alpha }_T={\alpha }_x$	${\text{β}}^{{\alpha }_x}$	${\alpha }_T=0$	${\text{β}}^0=1$
n (mol)	${\alpha }_n$	${\text{β}}^{{\alpha }_n}$	${\alpha }_n$	${\text{β}}^{{\alpha }_n}$	${\alpha }_n=3{\alpha }_x$	${\text{β}}^{3{\alpha }_x}$
$U_i$ (m ${\mathrm{s}}^{-1}$)	${\alpha }_U={\alpha }_x-{\alpha }_t$	${\text{β}}^{{\alpha }_x-{\alpha }_t}$	${\alpha }_U=0.5{\alpha }_x$	${\text{β}}^{0.5{\alpha }_x}$	${\alpha }_U=0$	${\text{β}}^0=1$
p (Pa)	${\alpha }_p=2{\alpha }_x-2{\alpha }_t+{\alpha }_{\rho }$	${\text{β}}^{2{\alpha }_x-2{\alpha }_t+{\alpha }_{\rho }}$	${\alpha }_p={\alpha }_x+{\alpha }_{\rho }$	${\text{β}}^{{\alpha }_x+{\alpha }_{\rho }}$	${\alpha }_p=0$	${\text{β}}^0=1$
$g_i$ ($\mathrm{m}{\mathrm{s}}^{-2}$)	${\alpha }_g={\alpha }_x-2{\alpha }_t$	${\text{β}}^{{\alpha }_x-2{\alpha }_t}$	${\alpha }_g=0$	${\text{β}}^0=1$	${\alpha }_g=-{\alpha }_x$	${\text{β}}^{-{\alpha }_x}$
μ (${\mathrm{m}}^2{\mathrm{s}}^{-1}$)	${\alpha }_{\nu }=2{\alpha }_x-{\alpha }_t$	${\text{β}}^{2{\alpha }_x-{\alpha }_t}$	${\alpha }_{\nu }=1.5{\alpha }_x$	${\text{β}}^{1.5{\alpha }_x}$	${\alpha }_{\nu }={\alpha }_x$	${\text{β}}^{{\alpha }_x}$
ν ($\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$)	${\alpha }_{\mu }=2{\alpha }_x+{\alpha }_{\rho }-{\alpha }_t$	${\text{β}}^{2{\alpha }_x+{\alpha }_{\rho }-{\alpha }_t}$	${\alpha }_{\mu }=1.5{\alpha }_x+{\alpha }_{\rho }$	${\text{β}}^{1.5{\alpha }_x+{\alpha }_{\rho }}$	${\alpha }_{\mu }={\alpha }_x$	${\text{β}}^{{\alpha }_x}$
σ (N ${\mathrm{m}}^{-1}$)	${\alpha }_{\sigma }=3{\alpha }_x-2{\alpha }_t+{\alpha }_{\rho }$	${\text{β}}^{3{\alpha }_x-2{\alpha }_t+{\alpha }_{\rho }}$	${\alpha }_{\sigma }=2{\alpha }_x+{\alpha }_{\rho }$	${\text{β}}^{2{\alpha }_x+{\alpha }_{\rho }}$	${\alpha }_{\sigma }={\alpha }_x$	${\text{β}}^{{\alpha }_x}$
κ (${\mathrm{m}}^{-1}$)	${\alpha }_{\kappa }={\alpha }_x^{-1}$	${\text{β}}^{{\alpha }_x^{-1}}$	${\alpha }_{\kappa }={\alpha }_x^{-1}$	${\text{β}}^{{\alpha }_x^{-1}}$	${\alpha }_{\kappa }={\alpha }_x^{-1}$	${\text{β}}^{{\alpha }_x^{-1}}$
${\nu }_t$ (${\mathrm{m}}^2{\mathrm{s}}^{-1}$)	${\alpha }_{{\nu }_t}=2{\alpha }_x-{\alpha }_t$	${\text{β}}^{2{\alpha }_x-{\alpha }_t}$	${\alpha }_{{\nu }_t}=1.5{\alpha }_x$	${\text{β}}^{1.5{\alpha }_x}$	${\alpha }_{{\nu }_t}={\alpha }_x$	${\text{β}}^{{\alpha }_x}$
$\overline{u_i{u}_j}$ (${\mathrm{m}}^2{\mathrm{s}}^{-2}$)	${\alpha }_{\overline{uu}}=2{\alpha }_x-2{\alpha }_t$	${\text{β}}^{2{\alpha }_x-2{\alpha }_t}$	${\alpha }_{\overline{uu}}={\alpha }_x$	${\text{β}}^{{\alpha }_x}$	${\alpha }_{\overline{uu}}=0$	${\text{β}}^0=1$
k (${\mathrm{m}}^2{\mathrm{s}}^{-2}$)	${\alpha }_k=2{\alpha }_x-2{\alpha }_t$	${\text{β}}^{2{\alpha }_x-2{\alpha }_t}$	${\alpha }_k={\alpha }_x$	${\text{β}}^{{\alpha }_x}$	${\alpha }_k=0$	${\text{β}}^0=1$
ϵ (${\mathrm{m}}^2{\mathrm{s}}^{-3}$)	${\alpha }_{ϵ}=2{\alpha }_x-3{\alpha }_t$	${\text{β}}^{2{\alpha }_x-3{\alpha }_t}$	${\alpha }_{ϵ}=0.5{\alpha }_x$	${\text{β}}^{0.5{\alpha }_x}$	${\alpha }_{ϵ}=-{\alpha }_x$	${\text{β}}^{-{\alpha }_x}$
$P_k$ (${\mathrm{m}}^2{\mathrm{s}}^{-3}$)	${\alpha }_{P_k}=2{\alpha }_x-3{\alpha }_t$	${\text{β}}^{2{\alpha }_x-3{\alpha }_t}$	${\alpha }_{P_k}=0.5{\alpha }_x$	${\text{β}}^{0.5{\alpha }_x}$	${\alpha }_{P_k}=-{\alpha }_x$	${\text{β}}^{-{\alpha }_x}$
R ($\mathrm{J}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$)	${\alpha }_R=2{\alpha }_x-2{\alpha }_t-{\alpha }_T$	${\text{β}}^{2{\alpha }_x-2{\alpha }_t-{\alpha }_T}$	${\alpha }_R=0$	${\text{β}}^0=1$	${\alpha }_R=0$	${\text{β}}^0=1$
$c_v$ ($\mathrm{J}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$)	${\alpha }_{c_v}=2{\alpha }_x-2{\alpha }_t-{\alpha }_T$	${\text{β}}^{2{\alpha }_x-2{\alpha }_t-{\alpha }_T}$	${\alpha }_{c_v}=0$	${\text{β}}^0=1$	${\alpha }_{c_v}=0$	${\text{β}}^0=1$
$M_m$ (kg ${\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$)	${\alpha }_{M_m}=3{\alpha }_x+{\alpha }_{\rho }-{\alpha }_n$	${\text{β}}^{3{\alpha }_x+{\alpha }_{\rho }-{\alpha }_n}$	${\alpha }_{M_m}=3{\alpha }_x+{\alpha }_{\rho }-{\alpha }_n$	${\text{β}}^{3{\alpha }_x+{\alpha }_{\rho }-{\alpha }_n}$	${\alpha }_{M_m}=0$	${\text{β}}^0=1$

Figure 1. Schematic illustration of the computational domain at prototype scale: (a) the longitudinal section of the domain, (b) a Taylor bubble region and (c) a cross-section of the mesh

Table 2. Scaling parameters and exponents used to scale the Taylor bubble prototype values to the corresponding values in the domains T4, T8 and T16 by using the NSLs

Domain	T4	T8	T16
Scaling parameter β	4	8	16
Scaling exponents
Length (m) ${\alpha }_x$	1	1	1
Time (s) ${\alpha }_t$	1	1	1
Density ($\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$) ${\alpha }_{\rho }$	0	0	0
Velocity (m ${\mathrm{s}}^{-1}$) ${\alpha }_U$	0	0	0
Gravitational acceleration (m ${\mathrm{s}}^{-2}$) ${\alpha }_g$	$-1$	$-1$	$-1$
Pressure (Pa) ${\alpha }_p$	0	0	0
Kinematic viscosity (${\mathrm{m}}^2{\mathrm{s}}^{-1}$) ${\alpha }_{\nu }$	1	1	1
Dynamic viscosity ($\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$) ${\alpha }_{\mu }$	1	1	1
Surface tension (N ${\mathrm{m}}^{-1}){\alpha }_{\sigma }$	1	1	1
Specific heat capacity (${\mathrm{m}}^2{\mathrm{K}}^{-1}{\mathrm{s}}^{-2}$) ${\alpha }_{c_v}$	0	0	0
Scaling ratios
Length (m) ${\text{β}}^{{\alpha }_x}$	4.0000	8.0000	16.0000
Time (s) ${\text{β}}^{{\alpha }_t}$	4.0000	8.0000	16.0000
Density ($\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$) ${\text{β}}^{{\alpha }_{\rho }}$	1.0000	1.0000	1.0000
Velocity ($\mathrm{m}{\mathrm{s}}^{-1}$) ${\text{β}}^{{\alpha }_U}$	1.0000	1.0000	1.0000
Gravitational acceleration (m ${\mathrm{s}}^{-2}$) ${\text{β}}^{{\alpha }_g}$	0.2500	0.1250	0.0625
Pressure (Pa) ${\text{β}}^{{\alpha }_p}$	1.0000	1.0000	1.0000
Kinematic viscosity (${\mathrm{m}}^2{\mathrm{s}}^{-1}$) ${\text{β}}^{{\alpha }_{\nu }}$	4.0000	8.0000	16.0000
Dynamic viscosity ($\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$) ${\text{β}}^{{\alpha }_{\mu }}$	4.0000	8.0000	16.0000
Surface tension (N ${\mathrm{m}}^{-1}$) ${\text{β}}^{{\alpha }_{\sigma }}$	4.0000	8.0000	16.0000
Specific heat capacity ($\mathrm{J}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$) ${\text{β}}^{{\alpha }_{c_v}}$	1.0000	1.0000	1.0000

Table 3. Parameters for the Taylor bubble in the prototype and the scaled domains

Domain	T1	T4	T8	T16	${\mathrm{T}4}_{\mathsf{\text{F}}}$	${\mathrm{T}8}_{\mathsf{\text{F}}}$	${\mathrm{T}16}_{\mathsf{\text{F}}}$
Diameter of the pipe d (m)	0.290000	0.072500	0.036250	0.018125	0.072500	0.036250	0.018125
Initial height of the bubble $h_b$ (m)	0.640	0.160	0.080	0.040	0.160	0.080	0.040
Initial height of the water plug (m)	3.360	0.840	0.420	0.210	0.840	0.420	0.210
Computational time (s)	3.50000	0.87500	0.43750	0.21875	1.75000	1.23700	0.87500
Gravitational acceleration g (m ${\mathrm{s}}^{-2}$)	9.81	39.24	78.48	156.96	9.81	9.81	9.81
Initial pressure inside the bubble $p_0$ (Pa)	134286.000	134286.000	134286.000	134286.000	33571.500	16785.750	8392.875
Density of water ${\rho }_w$ (kg ${\mathrm{m}}^{-3}$)	1000	1000	1000	1000	1000	1000	1000
Initial density of air ${\rho }_{a,0}$ (kg ${\mathrm{m}}^{-3}$)	1	1	1	1	1	1	1
Kinematic viscosity of water ${\nu }_w$ (${\mathrm{m}}^2{\mathrm{s}}^{-1}$)	$1\times {10}^{-6}$	$2.50\times {10}^{-7}$	$1.25\times {10}^{-7}$	$6.25\times {10}^{-8}$	$1\times {10}^{-6}$	$1\times {10}^{-6}$	$1\times {10}^{-6}$
Dynamic viscosity of water ${\mu }_w$ ($\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$)	$1\times {10}^{-3}$	$2.50\times {10}^{-4}$	$1.25\times {10}^{-4}$	$6.25\times {10}^{-5}$	$1\times {10}^{-3}$	$1\times {10}^{-3}$	$1\times {10}^{-3}$
Surface tension σ (N ${\mathrm{m}}^{-1}$)	0.07	$1.75\times {10}^{-2}$	$8.75\times {10}^{-3}$	$4.38\times {10}^{-3}$	0.07	0.07	0.07
Specific heat capacity of water $c_{v,w}$ ($\mathrm{J}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$)	4195	4195	4195	4195	4195	4195	4195
Specific heat capacity of air $c_{v,a}$ ($\mathrm{J}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$)	1007	1007	1007	1007	1007	1007	1007
Initial temperature $T_0$ (K)	293	293	293	293	293	293	293

Figure 2. Snapshot of the Taylor bubble at the cross-section $x_2^{\prime }=0$ and dimensionless time $t^{\prime }=2.90$ of (a) the prototype, (b) T16 and (c) ${\mathrm{T}16}_{\mathsf{\text{F}}}$

Figure 3. Snapshot of the Taylor bubble at the cross-section $x_2^{\prime }=0$ and dimensionless time $t^{\prime }=14.54$ of (a) the prototype, (b) T16 and (c) ${\mathrm{T}16}_{\mathsf{\text{F}}}$

Figure 4. Time history of the bubble nose location on the $x_3^{\prime }$ coordinate for the prototype and all the scaled domains

Table 4. $U_b^{\prime }$ for the prototype and all scaled domains

	T1	T4	T8	T16	${\mathrm{T}4}_{\mathsf{\text{F}}}$	${\mathrm{T}8}_{\mathsf{\text{F}}}$	${\mathrm{T}16}_{\mathsf{\text{F}}}$
$U_b^{\prime }$	0.486	0.476	0.495	0.482	0.398	0.363	0.337
$E_{U_b^{\prime }}$ (%)	–	2.06	1.85	0.82	18.11	25.31	30.66

Figure 5. Time histories of the averages of (a) the dimensionless density $\overline{{\rho }^{\prime }}$, (b) the dimensionless temperature $\overline{T^{\prime }}$ and (c) the dimensionless pressure $\overline{p^{\prime }}$ inside each modelled Taylor bubble and (d) the dimensionless volume $V^{\prime }$ for T1, T4, T8 and T16

Figure 6. Time histories of the averages of the (a) dimensionless density $\overline{{\rho }^{\prime }}$, (b) dimensionless temperature $\overline{T^{\prime }}$ and (c) dimensionless pressure $\overline{p^{\prime }}$ inside the bubbles and (d) the dimensionless volume $V^{\prime }$ for T1, ${\mathrm{T}4}_{\mathsf{\text{F}}}$, ${\mathrm{T}8}_{\mathsf{\text{F}}}$ and ${\mathrm{T}16}_{\mathsf{\text{F}}}$

Figure 7. Time history of the void fraction lost in the wake of the bubble for the prototype and the scaled domains