Numerical experiments of partial-depth colliding gravity currents using LES

Abstract

The collision of two opposing, horizontal gravity currents is investigated numerically using Large Eddy Simulation (LES). The classical lock-exchange configuration is considered for symmetric collision (currents with same densities and heights). A partial-depth setup is used which is considered to simulate better the collision in deep air column of atmosphere. LES results are validated using available experimental and DNS data (Fragoso et al. in J Fluid Mech 734:1–10, https://doi.org/10.1017/jfm.2013.475; Frantz et al. in Comput Fluids, 2021. https://doi.org/10.1016/j.compfluid.2021.104902) for the classical full depth lock-exchange gravity current. Numerical experiments are performed considering the effects of D/H (D is the height of dense fluid in the lock, H is the tank height) and Grashof number (Gr) on the collision characteristics. The former varies from 0.25 to 1.0 and the latter from $10^6$ to $10^{12}$. Maximum vertical displacement and maximum vertical velocity increase with decreasing D/H from 1.0 up to 0.5. They remain constant for $D/H \le 0.5$. Maximum vertical displacement decreases with Gr number opposite to the corresponding variation of the maximum equivalent height. Maximum vertical velocity decreases with increasing Gr number, due to increased turbulence. For $Gr \ge 5 \times 10^8$ (a) the maximum vertical displacement is almost constant (equal to 1.4D), (b) the temporal evolution of energies is approximately the same. At the time of maximum height the maximum potential energy and the minimum kinetic energy are approximately $80\%$ and $15\%$ of the initial potential energy, respectively, and (c) a region with intense turbulence and mixing of less dense fluid ($C\le 0.6$, C is the concentration) is formed after the occurrence of maximum height in the middle of the domain.

Appendix

To derive the energy budget the momentum equation (1b) is rewritten considering the material derivative for the terms on the LHS, the shear tensor ($\tau _{ij}$) for the diffusion term and the static pressure for the pressure term,

$$\begin{aligned} \begin{aligned} \frac{Du_i}{Dt}&=-\frac{1}{\rho _0}\frac{\partial }{\partial x_i}\left( p-\rho g_j x_j \right) + \frac{\partial \tau _{ij}}{\partial x_j}-g_jx_j\frac{\partial }{\partial x_i}(\frac{\rho }{\rho _0}) \Rightarrow \\ \frac{Du_i}{Dt}&=-\frac{1}{\rho _0}\frac{\partial p}{\partial x_i} + \frac{\partial \tau _{ij}}{\partial x_j}+ \frac{1}{\rho _0} \left( \frac{\partial (\rho g_j x_j)}{\partial x_i}-g_j x_j\frac{\partial \rho }{\partial x_i} \right) \Rightarrow \\ \frac{Du_i}{Dt}&=-\frac{1}{\rho _0}\frac{\partial p}{\partial x_i} + \frac{\partial \tau _{ij}}{\partial x_j}+ \frac{\rho }{\rho _0}g_i \end{aligned} \end{aligned}$$

(A.1)

Multiplying by $u_i$ and integrating over the whole domain V Eq. (A.1), reads

$$\begin{aligned} \begin{aligned}&u_i\frac{Du_i}{Dt}=-u_i\frac{1}{\rho _0}\frac{\partial p}{\partial x_i}+u_i\frac{\partial \tau _{ij}}{\partial x_j}+ u_i \left( \frac{\rho }{\rho _0} \right) g_i \Rightarrow \\&\frac{D}{Dt} \left( \frac{1}{2}u_iu_i \right) = - \frac{1}{\rho _0} \frac{\partial u_ip}{\partial x_i}+ u_i\frac{\partial \tau _{ij}}{\partial x_j}+\left( \frac{\rho }{\rho _0} \right) u_ig_i \Rightarrow \\&\int _V \frac{D}{Dt} \left( \frac{1}{2} u_iu_i \right) \,dV= -\int _V \frac{1}{\rho _0}\frac{\partial u_ip}{\partial x_i} \,dV +\int _V u_i \frac{\partial \tau _{ij}}{\partial x_j} \,dV +\int _V \left( \frac{\rho }{\rho _0} \right) g_i u_i \,dV \Rightarrow \\&\begin{aligned} \frac{D}{Dt} \left( \int _V \frac{1}{2} u_iu_i \right) \,dV&= - \frac{1}{\rho _0}\int _V \frac{\partial u_ip}{\partial x_i} \,dV +\int _V \frac{\partial u_i \tau _{ij}}{\partial x_j} \,dV - \int _V \tau _{ij} \frac{\partial u_i}{\partial x_j} \,dV \\&+\int _V \left( \frac{\rho }{\rho _0} \right) g_i u_i \,dV \end{aligned} \end{aligned} \end{aligned}$$

Considering the Gauss theorem the first and the second term on the RHS are equal to zero since the velocity components are zero on all the non-periodic boundaries. Hence,

$$\begin{aligned} \frac{dE_k}{dt}= - \int _V \tau _{ij} \frac{\partial u_i}{\partial x_j} \,dV +\int _V \left( \frac{\rho }{\rho _0} \right) g_i u_i \,dV \end{aligned}$$

(A.2)

Equation (3b) is differentiated considering that the effects of concentration diffusion on potential energy are negligible, which means $DC/Dt\approx 0$ or $D(\rho /\rho _0)/Dt\approx 0$. In addition, $g_iu_i=-|g |v$ where v is the vertical component of velocity. Hence,

$$\begin{aligned} \begin{aligned} \frac{dE_p}{dt}&= \int _V \left( \frac{\rho }{\rho _0} |g |\right) \frac{Dy}{Dt} \,dV + \int _V \frac{D}{Dt} \left( \frac{\rho }{\rho _0} \right) |g |y \,dV \\&=-\int _V \frac{\rho }{\rho _0} g_i u_i \,dV \end{aligned} \end{aligned}$$

(A.3)

Replacing Eq. (A.3) in Eq. (A.2) gives the Eq. (4) presented in Sect. 3.