This work deals with the modeling of the unsteady Newtonian fluid flow associated with an open cylindrical reservoir. This reservoir presents a hole on the right bottom wall. Fluid volume variation, heat and mass transfers are neglected. The unsteady governing equations are based on the conservation of mass and momentum. A finite volume technique is used to solve the non-dimensional equations and related boundary conditions. The algebraic system of equations resulting from the discretization process are solved by means of the THOMAS algorithm. For pressure-velocity coupling, the SIMPLE algorithm (Semi Implicit Method for Pressure Linked Equations) is used. Results for laminar flow (Re < 1000), including the pressure and velocities profiles as well as the streamlines in the reservoir are presented. Moreover, the effects of the D/d and H_{0}/D ratios and Reynolds number Re on the fluid flow are discussed. It is shown that the velocities and pressure depend essentially on the reservoir size. To validate the model, the present results have been compared with Zhou et al.’s results, Poiseuille’s and Bernoulli’s exact solution.
The reservoir has a very important role in industry applications. It is presented in different forms. It is considering here the open cylindrical form. The researchers generally focus their studies on the emptying of open cylindrical tank through an orifice centered on the bottom. In a drain, the fluid height decreases with the time and the orifice size [
The fluid has two types draining single-layer and double-layer. Zhou et al. [
To determine the time of fluid drain, Fadhilah et al. [
The numerical simulation of free surface flow in the reservoir has been studied by Xhang et al. [
Knowing the exact emptying time of vessel is very important because it gives an advantage to plan the work and permits to anticipate all the risks caused by the lost time. For this reason, Nasyrova et al. [
Peng et al. [
To be able to supply water drinkable in urban environment with a low investment cost, Camnasio et al. [
The purpose of Kalyan et al.’s [
The work of Memon et al. [
This article presents the concept of applying numerical model to study the unsteady 2D fluid flow in cylindrical reservoir with a hole based on the Navier-Stockes equations by analyzing the influences of the more important parameters on the velocities and pressure as the Reynolds number and the configuration parameters. The lack of work on the flow characterizing this configuration motivated us to carry out this research. In this work, governing equations are resolved using the finite volume method based on Thomas algorithm. Fluid volume variation, heat and mass transfer are neglected in this case. The tank configuration and flow parameters influence on the pressure and the velocity profiles as well as the streamlines profile in the tank are investigated.
The system in
Dimensionless continuity equation:
Dimensionless momentum equation:
Reynolds and Froude numbers:
Dimensionless boundary conditions:
At the reservoir outlet:
At the reservoir inlet:
On the walls:
The discretization of the dimensionless Navier Stokes equations as mentioned above using the finite volume method leads to the following general form [
With ϕ = (
The SIMPLE and Thomas algorithms have been used to solve respectively the pressure-velocity coupling and the algebraic equations system has been solved by THOMAS algorithm [
For the numerical simulation, the reservoir data illustrated in the
D/d | d/d | H_{0}/d | |
---|---|---|---|
4 | 1 | 15 | 10 |
Based on the analyzes of the influence of the meshes on the results, the mesh 70 × 30 nodes was chosen to save run time and memory for numerical computation.
The convergence criterion is determined such that the source term of pressure correction equation is less than 10^{−8}.
The evolution in z^{+ }= 7.5 of the adimensional velocity component v^{+} according to r^{+} describes a parabolic profile characteristic of a Poiseuille flow, see
By taking three values of t^{+}; z^{+} = 7.5; Fr = 0.1 and Re = 100, the dimensionless pressure profile as a function of r^{+} is given by
The Reynolds number Re is based on the average speed of water at the reservoir outlet. The increasing of the Reynolds number means that the velocity in the tank is intensifying.
For various Re,
The H_{0}/D ratio considerably influences the evolution in z^{+} = 7.5 of the speed component u^{+} according to radial component r^{+}, see
The D/d ratio considerably influences in z^{+} = 7.5 the evolution of adimensional speed component u^{+} according to r^{+} depicted in
To validate our model, the results of Zhou et al. [
The profile of curves on
By changing the diameter d of the orifice to the diameter D of the tank (d = D), our system can be assimilated as a vertical cylindrical pipe. This condition has been used to validate our model. The present result has been compared with Poiseuille’s result.
The one-dimensional unsteady Bernoulli equation was used to validate our results. By integrating this equation along the streamlines, the adimensional analytical expression is given by
To be able to compare the present result with that of Bernoulli, the orifice has been placed symmetrically at the reservoir’s bottom and it has been assumed that the flow is one-dimensional, that is to say that the variation of the speed is neglected along the direction r.
These two curves in
A 2D model has been developed to simulate the unsteady flow. The mathematical model is based on the mass conservation and momentum. It was assumed that there are no heat and mass transfers. The governing equations and the boundary conditions were discretized using the finite volume method. The solution of the algebraic equations system is assured by Thomas’s algorithm. The SIMPLE algorithm has been used to solve the pressure-velocity coupling. The results show us the influences of t^{+}, H_{0}/d, D/d, Re on the velocities (u^{+}, v^{+}) and the pressure. Comparisons between the present results and from those literatures were done and present a good agreement. The following conclusions are drawn:
(1) The existence of the orifice on the bottom wall causes the rapid increase (peak) of the adimensional velocity and the instability of the flow at the beginning. By moving away from the wall, the shear stress is less important and the fluid can flow freely. In accordance with the velocity u^{+}, the destabilization of the flow begins at the inlet of the hole and becomes more and more stable.
(2) The dimensionless velocities increase proportionally with the Reynolds number. For low Reynolds number, a parabolic profile of the boundary layer type was obtained. Otherwise, the regime tends towards a turbulent regime. The pressure always remains constant inside the tank. It is independent of time.
(3) The speed of ejection of the water at the outlet of the orifice increase proportionally with height of the water column. The velocity profile u^{+} continuously deforms along the height of the tank and tends towards a parabolic profile.
(4) The transfer of momentum by diffusion to the detriment of convection decrease with increase of the orifice diameter. The small value of orifice diameter causes the braking of the flow near the hole.
The continuity of this work concerns the experimental validation of the obtained theoretical results, the numerical study for turbulent or laminar flow in the same tank with or without obstacle by considering the heat and mass transfer at the reservoir wall.
Hole diameter [m]
Dimensionless hole diameter =
Reservoir diameter [m]
Froude number
Gravity acceleration [m.s^{−2}]
Reservoir height [m]
Total pressure [Pa]
Dimensionless pressure
Atmospheric pressure [Pa]
Radial coordinate [m]
Dimensionless radial coordinate =
Reynolds number =
Time [s]
Dimensionless time =
Maximal time [s]
Dimensionless maximal time
Average fluid velocity at outlet [m.s^{−1}]
Fluid velocity following
Dimensionless fluid velocity following z
Fluid velocity following
Dimensionless fluid velocity following
Axial coordinate [m]
Dimensionless axial coordinate =
Fluid density [kg.m^{−3}]
Dynamic viscosity of the fluid [kg.m^{−1}.s^{−1}]
Kinematic viscosity of the fluid [m^{2}.s^{−1}]
Thanks to the team of the Laboratory of Mechanics Energetic and Environment (LMEE) for their collaboration.