Axisymmetric gravity currents in a rotating system: experimental and numerical investigations

2001 ◽  
Vol 447 ◽  
pp. 1-29 ◽  
Author(s):  
MARK A. HALLWORTH ◽  
HERBERT E. HUPPERT ◽  
MARIUS UNGARISH
Keyword(s):  
Flow Field ◽  
Shallow Water ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Salt Water ◽  
Gravity Currents ◽  
The Body ◽  
Water Model ◽  
Maximum Radius ◽  
Theoretical Predictions

The propagation at high Reynolds number of a heavy, axisymmetric gravity current of given initial volume over a horizontal boundary is considered in both rotating and non-rotating situations. The investigation combines experiments with theoretical predictions by both shallow-water approximations and numerical solutions of the full axisymmetric equations. Attention is focused on cases when the initial ratio of Coriolis to inertia forces is small. The experiments were performed by quickly releasing a known cylindrical volume of dense salt water of 2 m diameter at the centre of a circular tank of diameter 13 m containing fresh ambient water of typical depth 80 cm. The propagation of the current was recorded for different initial values of the salt concentration, the volume of released fluid, the ratio of the initial height of the current to the ambient depth, and the rate of rotation. A major feature of the rotating currents was the attainment of a maximum radius of propagation. Thereafter a contraction–relaxation motion of the body of fluid and a regular series of outwardly propagating pulses was observed. The frequency of these pulses is slightly higher than inertial, and the amplitude is of the order of magnitude of half the maximum radius. Theoretical predictions of the corresponding gravity currents were also obtained by (i) previously developed shallow-water approximations (Ungarish & Huppert 1998) and (ii) a specially developed finite-difference code based on the full axisymmetric Navier–Stokes equations. The ‘numerical experiments’ provided by this code are needed to capture details of the flow field (such as the non-smooth shape of the interface, the vertical dependence of the velocity field) which are not reproduced by the shallow-water model and are very difficult for, or outside the range of, accurate experimental measurement. The comparisons and discussion provide insight into the flow field and indicate the advantages and limitations of the verified simulation tools.

scholarly journals Congested shallow water model: roof modeling in free surface flow

2018 ◽  
Vol 52 (5) ◽  
pp. 1679-1707 ◽  
Author(s):  
Edwige Godlewski ◽  
Martin Parisot ◽  
Jacques Sainte-Marie ◽  
Fabien Wahl
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Hyperbolic Equations ◽  
Mechanical Energy ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Water Model ◽  
Shallow Water Model ◽  
Time Step

We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

On the Flow Fields of Inertial Impactors

1974 ◽  
Vol 96 (4) ◽  
pp. 394-400 ◽  
Author(s):  
V. A. Marple ◽  
B. Y. H. Liu ◽  
K. T. Whitby
Keyword(s):  
Flow Field ◽  
Stokes Equations ◽  
Relaxation Method ◽  
Water Model ◽  
Navier Stokes ◽  
Visualization Technique ◽  
Navier Stokes Equations ◽  
Theoretical Results ◽  
Plate Distance ◽  
Good Agreement

The flow field in an inertial impactor was studied experimentally with a water model by means of a flow visualization technique. The influence of such parameters as Reynolds number and jet-to-plate distance on the flow field was determined. The Navier-Stokes equations describing the laminar flow field in the impactor were solved numerically by means of a finite difference relaxation method. The theoretical results were found to be in good agreement with the empirical observations made with the water model.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Numerical investigation of supercritical Taylor-vortex flow for a wide gap

1984 ◽  
Vol 138 ◽  
pp. 21-52 ◽  
Author(s):  
H. Fasel ◽  
O. Booz
Keyword(s):  
Flow Field ◽  
Vortex Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Taylor Number ◽  
Taylor Vortex ◽  
Plane Boundary ◽  
Aspect Ratios ◽  
Taylor Vortex Flow ◽  
Wide Gap

For a wide gap (R1/R2= 0.5) and large aspect ratiosL/d, axisymmetric Taylor-vortex flow has been observed in experiments up to very high supercritical Taylor (or Reynolds) numbers. This axisymmetric Taylor-vortex flow was investigated numerically by solving the Navier–Stokes equations using a very accurate (fourth-order in space) implicit finite-difference method. The high-order accuracy of the numerical method, in combination with large numbers of grid points used in the calculations, yielded accurate and reliable results for large supercritical Taylor numbers of up to 100Tac(or 10Rec). Prior to this study numerical solutions were reported up to only 16Tac. The emphasis of the present paper is placed upon displaying and elaborating the details of the flow field for large supercritical Taylor numbers. The flow field undergoes drastic changes as the Taylor number is increased from just supercritical to 100Tac. Spectral analysis (with respect toz) of the flow variables indicates that the number of harmonics contributing substantially to the total solution increases sharply when the Taylor number is raised. The number of relevant harmonics is already unexpectedly high at moderate supercriticalTa. For larger Taylor numbers, the evolution of a jetlike or shocklike flow structure can be observed. In the axial plane, boundary layers develop along the inner and outer cylinder walls while the flow in the core region of the Taylor cells behaves in an increasingly inviscid manner.

Models of non-Boussinesq lock-exchange flow

2011 ◽  
Vol 675 ◽  
pp. 1-26 ◽  
Author(s):  
R. ROTUNNO ◽  
J. B. KLEMP ◽  
G. H. BRYAN ◽  
D. J. MURAKI
Keyword(s):  
Free Boundary ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Water System ◽  
Analytic Solutions ◽  
Analytical Models ◽  
Navier Stokes ◽  
Exchange Flow

Nearly all analytical models of lock-exchange flow are based on the shallow-water approximation. Since the latter approximation fails at the leading edges of the mutually intruding fluids of lock-exchange flow, solutions to the shallow-water equations can be obtained only through the specification of front conditions. In the present paper, analytic solutions to the shallow-water equations for non-Boussinesq lock-exchange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for other proposed front conditions – conditions which appear to the shallow-water system as forced boundary conditions. Both solutions to the shallow-water equations are compared with the numerical solutions of the Navier–Stokes equations and a mixture of successes and failures is recorded. The apparent success of some aspects of the forced solutions of the shallow-water equations, together with the fact that in a real fluid the density interface is a free boundary, shows the need for an improved theory of lock-exchange flow taking into account non-hydrostatic effects for density interfaces intersecting rigid boundaries.

scholarly journals Towards a new friction model for shallow water equations through an interactive viscous layer

2019 ◽  
Vol 53 (1) ◽  
pp. 269-299 ◽  
Author(s):  
François James ◽  
Pierre-Yves Lagrée ◽  
Minh H. Le ◽  
Mathilde Legrand
Keyword(s):  
Shallow Water ◽  
Stokes Equations ◽  
Water Model ◽  
Finite Volume Scheme ◽  
Inviscid Fluid ◽  
Phase Lag ◽  
Viscous Layer ◽  
Viscous Effects ◽  
Shallow Water Models ◽  
Water Models

The derivation of shallow water models from Navier–Stokes equations is revisited yielding a class of two-layer shallow water models. An improved velocity profile is proposed, based on the superposition of an inviscid fluid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which depends not only on velocity and depth but also on the variations of velocity and thickness of the viscous layer. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the first two are mass and momentum conservation in which a slight correction on hydrostatic pressure has been made; the third one, known as von Kármán equation, describes the evolution of the viscous layer. This coupled model is shown to be conditionally hyperbolic, and a Godunov-type finite volume scheme is also proposed. Several numerical examples are provided and compared to the Multi-Layer Saint-Venant model. They emphasize the ability of the model to deal with unsteady viscous effects. They illustrate also the phase-lag between friction and topography, and even recover possible reverse flows.

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