2 results
Direct numerical simulation of stratified homogeneous turbulent shear flows
- T. Gerz, U. Schumann, S. E. Elghobashi
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- Journal:
- Journal of Fluid Mechanics / Volume 200 / March 1989
- Published online by Cambridge University Press:
- 26 April 2006, pp. 563-594
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The exact time-dependent three-dimensional Navier-Stokes and temperature equations are integrated numerically to simulate stably stratified homogeneous turbulent shear flows at moderate Reynolds numbers whose horizontal mean velocity and mean temperature have uniform vertical gradients. The method uses shear-periodic boundary conditions and a combination of finite-difference and pseudospectral approximations. The gradient Richardson number Ri is varied between 0 and 1. The simulations start from isotropic Gaussian fields for velocity and temperature both having the same variances.
The simulations represent approximately the conditions of the experiment by Komori et al. (1983) who studied stably stratified flows in a water channel (molecular Prandtl number Pr = 5). In these flows internal gravity waves build up, superposed by hot cells leading to a persistent counter-gradient heat-flux (CGHF) in the vertical direction, i.e. heat is transported from lower-temperature to higher-temperature regions. Further, simulations with Pr = 0.7 for air have been carried out in order to investigate the influence of the molecular Prandtl number. In these cases, no persistent CGHF occurred. This confirms our general conclusion that the counter-gradient heat flux develops for strongly stable flows (Ri ≈ 0.5–1.0) at sufficiently large Prandtl numbers (Pr = 5). The flux is carried by hot ascending, as well as cold descending turbulent cells which form at places where the highest positive and negative temperature fluctuations initially existed. Buoyancy forces suppress vertical motions so that the cells degenerate to two-dimensional fossil turbulence. The counter-gradient heat flux acts to enforce a quasi-static equilibrium between potential and kinetic energy.
Previously derived turbulence closure models for the pressure-strain and pressure-temperature gradients in the equations for the Reynolds stress and turbulent heat flux are tested for moderate-Reynolds-number flows with strongly stable stratification (Ri = 1). These models overestimate the turbulent interactions and underestimate the buoyancy contributions. The dissipative timescale ratio for stably stratified turbulence is a strong function of the Richardson number and is inversely proportional to the molecular Prandtl number of the fluid.
Direct numerical simulation of a three-dimensional spatially developing bubble-laden mixing layer with two-way coupling
- O. A. DRUZHININ, S. E. ELGHOBASHI
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- Journal:
- Journal of Fluid Mechanics / Volume 429 / 25 February 2001
- Published online by Cambridge University Press:
- 01 March 2001, pp. 23-61
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Direct numerical simulations (DNS) of a three-dimensional spatially-developing mixing layer (ML) laden with spherical gaseous bubbles are performed, with both one-way and two-way coupling between the two phases. Forcing is used to initialize the spanwise vortex roll-up and to create a pair of counter-rotating streamwise vortices, rendering the carrier flow three-dimensional. The characteristics of the resulting ML flow field are similar to those reported in numerous experimental and numerical studies. The volume fraction (or concentration) of the bubble phase is considered small enough to neglect bubble–bubble interactions. The no-slip fluid velocity condition is assumed at the bubble surface, and the bubble Reynolds number is less than 1 throughout the simulation time. The two-fluid formulation (TF) is used to compute the bubble-phase velocity and concentration and the two-way coupling source term in the fluid momentum equation. A Lagrangian–Eulerian mapping (LEM) solver is employed to solve the equations for the bubble velocity and concentration. LEM is capable of resolving the gradients of concentration created by the bubble preferential accumulation without numerical instabilities. Two different inflow profiles (Cref(z)) of bubble-phase concentration are considered: a uniform profile and a tanh-profile. In the latter case, the high-speed (upper) stream is devoid of bubbles, and the low-speed (lower) stream is uniformly laden with bubbles at the inflow plane.
The DNS results show that in addition to the well-known preferential accumulation of bubbles in the vortex centres, sheets of increased bubble concentration (C-sheets) develop in the rollers created by the vortex pairing in the ML core, with two local maxima of vorticity and an enhanced strain-rate field. The development of C-sheets is governed by the stretching and contraction along the principal axes of the local strain rate.
In the case of uniform Cref, the two-way coupling reduces the average ML vorticity thickness and the entrainment of the irrotational fluid into the ML core, as compared to the bubble-free case, upstream of the location of the first vortex pairing. However, both ML vorticity thickness is increased and entrainment is enhanced by the bubbles farther downstream, after the pairing. The fluid velocity fluctuations are reduced by the bubbles throughout the ML, as compared to the one-way coupling case.
In the case of the tanh-profile of Cref, the velocity fluctuations and the ML vorticity thickness are increased by the bubbles upstream the location of the first vortex pairing owing to the ‘unstable’ inflow bubble stratification (Druzhinin & Elghobashi 1998). On the other hand, the velocity fluctuations are reduced by the bubbles, and the ML vorticity thickness oscillates with the streamwise distance farther downstream.