Horizontal buoyant jet is a fundamental flow regime for hydrogen safety analysis in power industry. The purpose of this study is to develop a fast non-Boussinesq engineering model the horizontal buoyant round jets. Verification of this integral model is established with available experimental data and comparisons over a large range of density variations with the CFD codes GASFLOW. The model has proved to be an efficient engineering tool for predicting horizontal strongly buoyant round jets.

Turbulent buoyant jet is a fundamental flow regime in hydrogen safety analysis since it affects hydrogen distribution and mitigation measures when accidents occur [

Few experimental data and calculations on horizontal turbulent buoyant jet with large density variation can be found in the open literatures. Most of the experiments were carried out for the small density variation when the Boussinesq approximation is valid. Pantokratoras [

In this study non-Boussinesq integral model for horizontal buoyant round jet was derived with the modified entrainment hypothesis. The system of conservation equations of the integral model was solved by a forth order Runge-Kutta method to obtain numerical solutions in the transition region from jet-like to plume-like.

The problem description and modeling efforts are presented in Sections

The horizontal buoyant jet formed from a round orifice is discharged into the unbounded stagnant uniform ambient, as shown in Figure

Definition diagram for horizontal buoyant jet discharges from round orifice into the unstratified ambient.

In this study, the pressure across the flow is assumed to be uniform and equal to the ambient pressure outside of the boundary. The basic governing equations (neglecting the dissipation and turbulent transport in comparison with the mean flow) consist of mass, momentum, energy, and concentration conservation equations,

The divergence theorem is applied, and the basic governing equations become

A system of first-order ordinary differential equations was obtained after the integration, where the seven unknowns are the density, velocity, temperature along the trajectory,

The general assumptions made in this investigation are as follows.

The flow is fully turbulent which means there is no Reynold number dependence.

The profiles of velocity, density, and temperature are similar at all cross-sections normal to the jet trajectory.

Longitudinal turbulent transport is small compared with latitudinal convective transport.

Velocity profile is assumed to be Gaussian distribution:

Density deficiency profile with respect to the ambient density in a uniform ambient is assumed to be Gaussian:^{2} is the turbulent Schmidt number, which is assumed to be constant and is usually found to be somewhat larger than 1 for small density ratio cases. In this study

Due to the large density or temperature variation between the jet and the ambient considered in the non-Boussinesq model, the density in the trajectory

To close the equations system, the mass entrainment rate should be specified. The entrainment relation for the horizontal round jet is given by:

For low-momentum buoyant jets, experimental data indicates that the local rate of entrainment increases as the jets leaves the momentum-dominated region and enters a region where the effects of buoyancy become more pronounced. In Jirka’s paper [

List summarized much of the work on the entrainment hypothesis and proposed values of

In the non-Boussinesq model, the effect of large density variation should be considered in the entrainment coefficient. In this study the local entrainment coefficient for the horizontal buoyant jet is assumed as:

For the pure jet

Centerline velocity decay for pure jets.

Concentration decay along the centerline for pure jets.

The horizontal buoyant jets with small density variations (

Normalized trajectories of horizontal buoyant jet.

The length scale

The normalized centerline dilution

Normalized centerline dilutions of horizontal buoyant jet.

The predictions of non-Boussinsq model agree well with the experimental data. When the initial Froude number

The non-Boussinesq integral model provides a satisfactory transition behavior for the horizontal buoyant jets with small density variations from the jet-like to plume-like region. Figures

Velocity decay of horizontal buoyant jet.

Froude number decay of horizontal buoyant jet.

Entrainment coefficients of horizontal buoyant jet.

The horizontal buoyant jets with large density variations, for instance hydrogen or helium injecting into air, have not received sufficient research before, and almost no experimental data could be found in the open literature. CFD code GASFLOW [

Figure

Trajectories of horizontal buoyant jet with large density variations.

Velocity decay of horizontal buoyant jet with large density variations.

Concentration decay of horizontal buoyant jet with large density variations.

Before the Gaussian profiles are reached, the initial unsheared profiles undergo changes in form of peripherally growing axis symmetric mixing layers. This initial region is called the zone of flow establishment which lacks of self-similarity. The transition in this region is complex and rapid, and the distance is up to 5–10 diameter of the orifice. A distance of 5–10 diameters from the orifice is shifted in the study.

The mass entrainment coefficients in this study were obtained under the experimental conditions when the velocity and density variation are not so high. How the high velocity and large density variation affect the entrainment coefficient is not clear. In the recent simulation of the underexpanded hydrogen jet [

The mechanisms of these uncertainties needs further study in the future work.

This non-Boussinesq integral model developed in the study is a fast engineering model to solve the horizontal buoyant round jets problems. The model was validated by the pure jet, horizontal buoyant jets with small/large density variations, and good agreements with the experimental data, and CFD predictions were obtained.

For strongly buoyant jet the Boussinesq approximation is violated which will over-predict the mass entrainment and under-estimate the buoyancy effect [

The entrainment assumption is a key requirement for the integral model. The entrainment assumption taking into account the Richard number and the angle