scholarly journals Numerical validation of novel scaling laws for air entrainment in water

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Daniele Catucci ◽  
Riccardo Briganti ◽  
Valentin Heller
Keyword(s):  
Scaling Laws ◽  
Stokes Equations ◽  
Scale Effects ◽  
Air Entrainment ◽  
Navier Stokes ◽  
The Novel ◽  
Water Flows ◽  
Wide Range ◽  
Self Similar ◽  
Flow Variables

The Froude scaling laws have been used to model a wide range of water flows at reduced size for almost a century. In such Froude scale models, significant scale effects for air–water flows (e.g. hydraulic jumps or wave breaking) are typically observed. This study introduces novel scaling laws, excluding scale effects in the modelling of air–water flows. This is achieved by deriving the conditions under which the governing equations are self-similar. The one-parameter Lie group of point-scaling transformations is applied to the Reynolds-averaged Navier–Stokes equations, including surface tension effects. The scaling relationships between variables are derived for the flow variables, fluid properties and initial and boundary conditions. Numerical simulations are conducted to validate the novel scaling laws for (i) a dam break flow interacting with an obstacle and (ii) a vertical plunging water jet. Results for flow variables, void fraction and turbulent kinetic energy are shown to be self-similar at different scales, i.e. they collapse in dimensionless form. Moreover, these results are compared with those obtained using the traditional Froude scaling laws, showing significant scale effects. The novel scaling laws are a more universal and flexible alternative with a genuine potential to improve laboratory modelling of air–water flows.

Self-similar behaviour of a rotor wake vortex core

2014 ◽  
Vol 740 ◽  
Author(s):  
Mohamed Ali ◽  
Malek Abid
Keyword(s):  
Vortex Core ◽  
Scaling Laws ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Azimuthal Velocity ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Rotating Blade ◽  
Self Similar

AbstractWe report a self-similar behaviour of solutions (obtained numerically) of the Navier–Stokes equations behind a single-blade rotor. That is, the helical vortex core in the wake of a rotating blade is self-similar as a function of its age. Profiles of vorticity and azimuthal velocity in the vortex core are characterized, their similarity variables are identified and scaling laws of these variables are given. Solutions of incompressible three-dimensional Navier–Stokes equations for Reynolds numbers up to $Re= 2000$ are considered.

Stern Flows at Full-Scale Reynolds Numbers

1991 ◽  
Vol 35 (02) ◽  
pp. 101-113
Author(s):  
S. Ju ◽  
V. C. Patel
Keyword(s):  
Scaling Laws ◽  
Stokes Equations ◽  
Scale Effects ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Axisymmetric Body ◽  
Navier Stokes ◽  
Wall Functions ◽  
Ship Wake ◽  
High Reynolds

A numerical method for the solution of the Reynolds-averaged Navier-Stokes equations for three-dimensional flow of an incompressible fluid, used previously for calculations at model-scale Reynolds numbers, has been applied to the flow over the stern of an axisymmetric body and a representative ship over a range of Reynolds numbers up to five billion. High Reynolds number experiments on axi-symmetric bodies strongly support the use of wall functions in the turbulence model. Comparisons with traditional ship wake scaling laws suggest that such laws do not properly describe the scale effects on the flow in the propeller plane.

GLOBAL EXISTENCE TO THE NAVIER–STOKES EQUATIONS THROUGH A SELF-SIMILAR-LIKE UPPER SOLUTION

2012 ◽  
Vol 14 (05) ◽  
pp. 1250031
Author(s):  
GUY BERNARD
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Initial Velocity ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Navier Stokes Equations ◽  
Upper Solution ◽  
Self Similar

A global existence result is presented for the Navier–Stokes equations filling out all of three-dimensional Euclidean space ℝ3. The initial velocity is required to have a bell-like form. The method of proof is based on symmetry transformations of the Navier–Stokes equations and a specific upper solution to the heat equation in ℝ3× [0, 1]. This upper solution has a self-similar-like form and models the diffusion process of the heat equation. By a symmetry transformation, the problem is transformed into an equivalent one having a very small initial velocity. Using the upper solution, the equivalent problem is then solved in the time interval [0, 1]. This local solution is then extended to the time interval [0, ∞) by an iterative process. At each step, the problem is extended further in time in an interval of time whose length is greater than one, thus producing the global solution. Each extension is transformed, by an appropriate change of variables, into the first local problem in the time interval [0, 1]. These transformations exploit the diffusive and self-similar-like nature of the upper solution.

Method for Non-Dimensional Tip Leakage Flow Characterization in Radial Turbines

2018 ◽  
Author(s):  
José Ramón Serrano ◽  
Roberto Navarro ◽  
Luis Miguel García-Cuevas ◽  
Lukas Benjamin Inhestern
Keyword(s):  
Suction Side ◽  
Tip Clearance ◽  
Navier Stokes ◽  
The Novel ◽  
Tip Leakage Flow ◽  
Navier Stokes Equation ◽  
Tip Clearance Flow ◽  
Tip Leakage ◽  
Wide Range ◽  
Clearance Flow

Tip leakage loss characterization and modeling plays an important role in small size radial turbine research. The momentum of the flow passing through the tip gap is highly related with the tip leakage losses. The ratio of fluid momentum driven by the pressure gradient between suction side and pressure side and the fluid momentum caused by the shroud friction has been widely used to analyze and to compare different sized tip clearances. However, the commonly used number for building this momentum ratio lacks some variables, as the blade tip geometry data and the viscosity of the used fluid. To allow the comparison between different sized turbocharger turbine tip gaps, work has been put into finding a consistent characterization of radial tip clearance flow. Therefore, a non-dimensional number has been derived from the Navier Stokes Equation. This number can be calculated like the original ratio over the chord length. Using the results of wide range CFD data, the novel tip leakage number has been compared with the traditional and widely used ratio. Furthermore, the novel tip leakage number can be separated into three different non-dimensional factors. First, a factor dependent on the radial dimensions of the tip gap has been found. Second, a factor defined by the viscosity, the blade loading, and the tip width has been identified. Finally, a factor that defines the coupling between both flow phenomena. These factors can further be used to filter the tip gap flow, obtained by CFD, with the influence of friction driven and pressure driven momentum flow.

Effects of Buoyancy on Laminar Flow in Curved Elliptic Ducts

1992 ◽  
Vol 114 (4) ◽  
pp. 936-943 ◽  
Author(s):  
Z. F. Dong ◽  
M. A. Ebadian
Keyword(s):  
Laminar Flow ◽  
Stokes Equations ◽  
Control Volume ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Uniform Heat Flux ◽  
Published Data ◽  
Circular Duct ◽  
Uniform Wall Temperature ◽  
Wide Range

This paper numerically investigates the effects of buoyancy on fully developed laminar flow in a curved duct with an elliptic cross section. The flow of Newtonian fluids is assumed steady in terms of Boussinesq approximation. The curved elliptic duct is subjected to thermal boundary conditions of axially uniform heat flux and peripherally uniform wall temperature. The numerically generated boundary-fitted coordinate system is applied to discretize the solution domain of the elliptic duct, and the Navier-Stokes equations and the energy equation, including the curvature ratio, are solved by use of the control volume-based finite difference method. The solution covers a wide range of curvature ratios, and Dean and Grashof numbers. The results presented are displayed graphically and in tabular form to illustrate the buoyancy effect. It is further shown that buoyancy acts to increase both the Nusselt number and the friction factor and changes the distribution of the velocity and the temperature. The results for the curved circular duct with and without buoyancy are compared with the data available in the open literature for all cases. Also compared with the published data are the results of laminar flow in a curved elliptic duct, and very good agreement is obtained.

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

scholarly journals Gyrotactic swimmers in turbulence: shape effects and role of the large-scale flow

2018 ◽  
Vol 856 ◽  
Author(s):  
M. Borgnino ◽  
G. Boffetta ◽  
F. De Lillo ◽  
M. Cencini
Keyword(s):  
Large Scale ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Small Scale ◽  
Navier Stokes Equations ◽  
Preferential Sampling ◽  
Wide Range ◽  
Shape Effects ◽  
Dilute Suspensions ◽  
Large Scale Flow

We study the dynamics and the statistics of dilute suspensions of gyrotactic swimmers, a model for many aquatic motile microorganisms. By means of extensive numerical simulations of the Navier–Stokes equations at different Reynolds numbers, we investigate preferential sampling and small-scale clustering as a function of the swimming (stability and speed) and shape parameters, considering in particular the limits of spherical and rod-like particles. While spherical swimmers preferentially sample local downwelling flow, for elongated swimmers we observe a transition from downwelling to upwelling regions at sufficiently high swimming speed. The spatial distribution of both spherical and elongated swimmers is found to be fractal at small scales in a wide range of swimming parameters. The direct comparison between the different shapes shows that spherical swimmers are more clusterized at small stability and speed numbers, while for large values of the parameters elongated cells concentrate more. The relevance of our results for phytoplankton swimming in the ocean is briefly discussed.

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