scholarly journals Models of internal jumps and the fronts of gravity currents: unifying two-layer theories and deriving new results

2018 ◽  
Vol 846 ◽  
pp. 654-685 ◽  
Author(s):  
Marius Ungarish ◽  
Andrew J. Hogg
Keyword(s):  
Velocity Field ◽  
Control Volume ◽  
Gravity Current ◽  
Vortex Sheet ◽  
Gravitational Acceleration ◽  
Two Fluids ◽  
Dissipative Effects ◽  
Wake Model ◽  
New Formulation ◽  
Vorticity Balance

The steady speeds of the front of a gravity current and of an internal jump on a two-layer stratification are often sought in terms of the heights of the relatively dense fluid both up- and downstream from the front or jump, the height of the channel within which they flow, the densities of the two fluids and gravitational acceleration. In this study a unifying framework is presented for calculating the speeds by balancing mass and momentum fluxes across a control volume spanning the front or jump and by ensuring the assumed pressure field is single-valued, which is shown to be equivalent to forming a vorticity balance over the control volume. Previous models have assumed the velocity field is piecewise constant in each layer with a vortex sheet at their interface and invoked explicit or implicit closure assumptions about the dissipative effects to derive the speed. The new formulation yields all of the previously presented expressions and demonstrates that analysing the vorticity balance within the control volume is a useful means of constraining possible closure assumptions, which is arguably more effective than consideration of the flow energetics. However the new approach also reveals that a novel class of models may be developed in which there is shear in the velocity field in the wake downstream of the front or the jump, thus spreading the vorticity over a layer of non-vanishing thickness, rather than concentrating it into a vortex sheet. Mass, momentum and vorticity balances applied over the control volume allow the thickness of the wake and the speed of the front/jump to be evaluated. Results from this vortex-wake model are consistent with published numerical simulations and with data from laboratory experiments, and improve upon predictions from previous formulae. The results may be applied readily to Boussinesq and non-Boussinesq systems and because they arise as simple algebraic expressions, can be straightforwardly incorporated as jump conditions into spatially and temporally varying descriptions of the motion.

Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods

1978 ◽  
Vol 84 (3) ◽  
pp. 433-453 ◽  
Author(s):  
J. C. S. Meng ◽  
J. A. L. Thomson
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Function Method ◽  
Gravity Current ◽  
Vortex Sheet ◽  
Taylor Instability ◽  
Green’S Function Method ◽  
Discrete Vortex ◽  
Two Fluids ◽  
Rayleigh Taylor Instability

A class of nonlinear hydrodynamic problems is studied. Physical problems such as shear flow, flow with a sharp interface separating two fluids of different density and flow in a porous medium all belong to this class. Owing to the density difference across the interface, vorticity is generated along it by the interaction between the gravitational pressure gradient and the density gradient, and the motion consists of essentially two processes: the creation of a vortex sheet and the subsequent mutual induction of different portions of this sheet.Two numerical methods are investigated. One is based upon the well-known Green's function method, which is a Lagrangian method using the Biot-Savart law, while the other is the vortex-in-cell (VIC) method, which is a Lagrangian-Eulerian method. Both methods treat the interface as sharp and represent it by a distribution of point vortices. The VIC method applies the FFT (fast Fourier transform) to solve the stream-function/vorticity equation on an Eulerian grid, and computational efficiency is further improved by using the reality properties of the physical variables.Four specific problems are investigated numerically in this paper. They are: the Rayleigh-Taylor instability, the Saffman-Taylor instability, transport of aircraft trailing vortices in a wind shear, and the gravity current. All four problems are solved using the VIC method and the results agree well with results obtained by previous investigators. The first two problems, the Rayleigh-Taylor instability and the Saffman-Taylor instability, are also solved by the Green's function method. Comparisons of results obtained by the two methods show good agreement, but, owing to its computational economy, the VIC method is concluded to be the better method for treating the class of hydrodynamic problems considered here.

scholarly journals A New Formulation of Maxwell’s Equations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 868
Author(s):  
Simona Fialová ◽  
František Pochylý
Keyword(s):  
Numerical Methods ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Control Volume ◽  
Control Volume Method ◽  
Integral Forms ◽  
New Formulation ◽  
Electrical Induction ◽  
Volume Method ◽  
Integral Identities

In this paper, new forms of Maxwell’s equations in vector and scalar variants are presented. The new forms are based on the use of Gauss’s theorem for magnetic induction and electrical induction. The equations are formulated in both differential and integral forms. In particular, the new forms of the equations relate to the non-stationary expressions and their integral identities. The indicated methodology enables a thorough analysis of non-stationary boundary conditions on the behavior of electromagnetic fields in multiple continuous regions. It can be used both for qualitative analysis and in numerical methods (control volume method) and optimization. The last Section introduces an application to equations of magnetic fluid in both differential and integral forms.

Tracing Streamlines on Unstructured Grids From Finite Volume Discretizations

SPE Journal ◽  
2008 ◽  
Vol 13 (04) ◽  
pp. 423-431 ◽  
Author(s):  
Sebastien F. Matringe ◽  
Ruben Juanes ◽  
Hamdi A. Tchelepi
Keyword(s):  
Finite Element ◽  
Velocity Field ◽  
Finite Difference ◽  
Finite Volume ◽  
Unstructured Grids ◽  
Control Volume ◽  
Next Generation ◽  
Flux Approximation ◽  
Streamline Tracing ◽  
Tracing Method

Summary The accuracy of streamline reservoir simulations depends strongly on the quality of the velocity field and the accuracy of the streamline tracing method. For problems described on complex grids (e.g., corner-point geometry or fully unstructured grids) with full-tensor permeabilities, advanced discretization methods, such as the family of multipoint flux approximation (MPFA) schemes, are necessary to obtain an accurate representation of the fluxes across control volume faces. These fluxes are then interpolated to define the velocity field within each control volume, which is then used to trace the streamlines. Existing methods for the interpolation of the velocity field and integration of the streamlines do not preserve the accuracy of the fluxes computed by MPFA discretizations. Here we propose a method for the reconstruction of the velocity field with high-order accuracy from the fluxes provided by MPFA discretization schemes. This reconstruction relies on a correspondence between the MPFA fluxes and the degrees of freedom of a mixed finite-element method (MFEM) based on the first-order Brezzi-Douglas-Marini space. This link between the finite-volume and finite-element methods allows the use of flux reconstruction and streamline tracing techniques developed previously by the authors for mixed finite elements. After a detailed description of our streamline tracing method, we study its accuracy and efficiency using challenging test cases. Introduction The next-generation reservoir simulators will be unstructured. Several research groups throughout the industry are now developing a new breed of reservoir simulators to replace the current industry standards. One of the main advances offered by these next generation simulators is their ability to support unstructured or, at least, strongly distorted grids populated with full-tensor permeabilities. The constant evolution of reservoir modeling techniques provides an increasingly realistic description of the geological features of petroleum reservoirs. To discretize the complex geometries of geocellular models, unstructured grids seem to be a natural choice. Their inherent flexibility permits the precise description of faults, flow barriers, trapping structures, etc. Obtaining a similar accuracy with more traditional structured grids, if at all possible, would require an overwhelming number of gridblocks. However, the added flexibility of unstructured grids comes with a cost. To accurately resolve the full-tensor permeabilities or the grid distortion, a two-point flux approximation (TPFA) approach, such as that of classical finite difference (FD) methods is not sufficient. The size of the discretization stencil needs to be increased to include more pressure points in the computation of the fluxes through control volume edges. To this end, multipoint flux approximation (MPFA) methods have been developed and applied quite successfully (Aavatsmark et al. 1996; Verma and Aziz 1997; Edwards and Rogers 1998; Aavatsmark et al. 1998b; Aavatsmark et al. 1998c; Aavatsmark et al. 1998a; Edwards 2002; Lee et al. 2002a; Lee et al. 2002b). In this paper, we interpret finite volume discretizations as MFEM for which streamline tracing methods have already been developed (Matringe et al. 2006; Matringe et al. 2007b; Juanes and Matringe In Press). This approach provides a natural way of reconstructing velocity fields from TPFA or MPFA fluxes. For finite difference or TPFA discretizations, the proposed interpretation provides mathematical justification for Pollock's method (Pollock 1988) and some of its extensions to distorted grids (Cordes and Kinzelbach 1992; Prévost et al. 2002; Hægland et al. 2007; Jimenez et al. 2007). For MPFA, our approach provides a high-order streamline tracing algorithm that takes full advantage of the flux information from the MPFA discretization.

On symmetric intrusions in a linearly stratified ambient: a revisit of Benjamin's steady-state propagation results

2021 ◽  
Vol 929 ◽  
Author(s):  
M. Ungarish
Keyword(s):  
Steady State ◽  
Control Volume ◽  
Gravity Current ◽  
Head Loss ◽  
Stability Criteria ◽  
Ambient Fluid ◽  
Height Ratio ◽  
Stagnation Line ◽  
Volume Analysis ◽  
State Propagation

Previous studies have extended Benjamin's theory for an inertial steady-state gravity current of density $\rho _{c}$ in a homogeneous ambient fluid of density $\rho _{o} < \rho _{c}$ to the counterpart propagation in a linearly stratified (Boussinesq) ambient (density decreases from $\rho _b$ to $\rho _{o}$ ). The extension is typified by the parameter $S = (\rho _{b}-\rho _{o})/(\rho _{c}-\rho _{o}) \in (0,1]$ , uses Long's solution for the flow over a topography to model the flow of the ambient over the gravity current, and reduces well to the classical theory for small and moderate values of $S$ . However, for $S=1$ , i.e. $\rho _b = \rho _c$ , which corresponds to a symmetric intrusion, various idiosyncrasies appear. Here attention is focused on this case. The control-volume analysis (balance of volume, mass, momentum and vorticity) produces a fairly compact analytical formulation, pending a closure for the head loss, and subject to stability criteria (no inverse stratification downstream). However, we show that plausible closures that work well for the non-stratified current (like zero head loss on the stagnation line, or zero vorticity diffusion) do not produce satisfactory results for the intrusion (except for some small ranges of the height ratio of current to channel, $a = h/H$ ). The reasons and insights are discussed. Accurate data needed for comparison with the theoretical model are scarce, and a message of this paper is that dedicated experiments and simulations are needed for the clarification and improvement of the theory.

An Assignment Procedure from Particles to Mesh that Preserves Field Values

2019 ◽  
Vol 17 (02) ◽  
pp. 1850130 ◽  
Author(s):  
Daniel Duque ◽  
Pep Español
Keyword(s):  
Fluid Dynamics ◽  
Finite Element Method ◽  
Computational Fluid Dynamics ◽  
Finite Element ◽  
Velocity Field ◽  
Cfd Simulation ◽  
Vortex Sheet ◽  
The Finite Element Method ◽  
Particle In Cell ◽  
Field Information

In computational fluid dynamics there have been many attempts to combine the advantages of having a fixed mesh, on which to carry out spatial calculations, with using particles moving according to the velocity field. These ideas in fact go back to particle-in-cell methods, proposed about 60 years ago. Of course, some procedure is needed to transfer field information between particles and mesh. There are many possible choices for this “assignment”, or “projection”. Several requirements may guide this choice. Two well-known ones are conservativity and stability, which apply to volume integrals of the fields. An additional one is here considered: preservation of information. This means that assignment from the particles onto the mesh and back should yield the same field values when the particles and the mesh coincide in position. The resulting method is termed “mass” assignment, due to its strong similarities with the finite element method. Several procedures are tested, including the well-known FLIP, on three scenarios: simple 1D convection, 2D convection of Zalesak’s disk, and a CFD simulation of the Taylor–Green periodic vortex sheet. Mass assignment is seen to be clearly superior to other methods.

A computational model of the flight dynamics and aerodynamics of a jellyfish-like flying machine

2017 ◽  
Vol 819 ◽  
pp. 621-655 ◽  
Author(s):  
Fang Fang ◽  
Kenneth L. Ho ◽  
Leif Ristroph ◽  
Michael J. Shelley
Keyword(s):  
Aerodynamic Force ◽  
Fast Multipole Method ◽  
Flow Simulation ◽  
Ground Effect ◽  
Vortex Sheet ◽  
Mirror Image ◽  
Design Parameters ◽  
Efficiency Increase ◽  
Wake Model ◽  
Opening And Closing

We explore theoretically the aerodynamics of a recently fabricated jellyfish-like flying machine (Ristroph & Childress, J. R. Soc. Interface, vol. 11 (92), 2014, 20130992). This experimental device achieves flight and hovering by opening and closing opposing sets of wings. It displays orientational or postural flight stability without additional control surfaces or feedback control. Our model ‘machine’ consists of two mirror-symmetric massless flapping wings connected to a volumeless body with mass and moment of inertia. A vortex sheet shedding and wake model is used for the flow simulation. Use of the fast multipole method allows us to simulate for long times and resolve complex wakes. We use our model to explore the design parameters that maintain body hovering and ascent, and investigate the performance of steady ascent states. We find that ascent speed and efficiency increase as the wings are brought closer, due to a mirror-image ‘ground-effect’ between the wings. Steady ascent is approached exponentially in time, which suggests a linear relationship between the aerodynamic force and ascent speed. We investigate the orientational stability of hovering and ascent states by examining the flyer’s free response to perturbation from a transitory external torque. Our results show that bottom-heavy flyers (centre of mass below the geometric centre) are capable of recovering from large tilts, whereas the orientation of the top-heavy flyers diverges. These results are consistent with the experimental observations in Ristroph & Childress (J. R. Soc. Interface, vol. 11 (92), 2014, 20130992), and shed light upon future designs of flapping-wing micro aerial vehicles that use jet-based mechanisms.

A laboratory study of the velocity structure in an intrusive gravity current

2002 ◽  
Vol 456 ◽  
pp. 33-48 ◽  
Author(s):  
RYAN J. LOWE ◽  
P. F. LINDEN ◽  
JAMES W. ROTTMAN
Keyword(s):  
Velocity Field ◽  
Velocity Structure ◽  
Fluid Velocity ◽  
Gravity Current ◽  
Maximum Speed ◽  
Wake Region ◽  
Tail Region ◽  
Front Speed ◽  
Energy Conserving ◽  
Lock Gate

Laboratory experiments were performed in which an intrusive gravity current was observed using shadowgraph and particle tracking methods. The intrusion was generated in a two-layer fluid with a sharp interface by mixing the fluid behind a vertical lock gate and then suddenly withdrawing the gate from the tank. The purpose of the experiments was to determine the structure of the velocity field inside the intrusion and the stability characteristics of the interface. Soon after the removal of the lock gate, the front of the intrusive gravity current travelled at a constant speed close to the value predicted by theory for an energy-conserving gravity current. The observed structure of the flow inside the intrusion can be divided into three regions. At the front of the intrusion there is an energy-conserving head region in which the fluid velocity is nearly uniform with speed equal to the front speed. This is followed by a dissipative wake region in which large billows are present with their associated mixing and in which the fluid velocity is observed to be non-uniform and have a maximum speed approximately 50% greater than the front speed. Behind the wake region is a tail region in which there is very little mixing and the velocity field is nearly uniform with a speed slightly faster than the front speed.

Numerical and experimental analysis of two-dimensional separated flows over a flexible sail

2002 ◽  
Vol 466 ◽  
pp. 319-341 ◽  
Author(s):  
O. LORILLU ◽  
R. WEBER ◽  
J. HUREAU
Keyword(s):  
Velocity Field ◽  
Numerical Data ◽  
Separated Flows ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Two Dimensional Model ◽  
Wake Model ◽  
Lift And Drag ◽  
Particle Imaging ◽  
Good Agreement

This paper is a numerical analysis of the flow over a exible sail with the usual two-dimensional model of ideal weightless incompressible fluid. The sail is assumed to be impervious, inelastic and weightless, and may or may not be mounted on a mast. Separated or attached flows are considered at any angle of attack. Our method is validated by numerical and experimental results, i.e. the sail shape and velocity field are determined by particle imaging velocimetry, and lift and drag by aerodynamic balance. Despite the simplicity of the wake model we use (the Helmholtz model), the computed free streamline geometry and especially the sail shape are in good agreement with the experimental and numerical data.

Optimal flexibility of a flapping appendage in an inviscid fluid

2008 ◽  
Vol 614 ◽  
pp. 355-380 ◽  
Author(s):  
SILAS ALBEN
Keyword(s):  
Power Law ◽  
Leading Edge ◽  
Vortex Sheet ◽  
Input Power ◽  
The Body ◽  
Flexible Body ◽  
Inviscid Fluid ◽  
Driving Frequency ◽  
Pitch Frequency ◽  
New Formulation

We present a new formulation of the motion of a flexible body with a vortex-sheet wake and use it to study propulsive forces generated by a flexible body pitched periodically at the leading edge in the small-amplitude regime. We find that the thrust power generated by the body has a series of resonant peaks with respect to rigidity, the highest of which corresponds to a body flexed upwards at the trailing edge in an approximately one-quarter-wavelength mode of deflection. The optimal efficiency approaches 1 as rigidity becomes small and decreases to 30–50% (depending on pitch frequency) as rigidity becomes large. The optimal rigidity for thrust power increases from approximately 60 for large pitching frequency to ∞ for pitching frequency 0.27. Subsequent peaks in response have power-law scalings with respect to rigidity and correspond to higher-wavenumber modes of the body. We derive the power-law scalings by analysing the fin as a damped resonant system. In the limit of small driving frequency, solutions are self-similar at the leading edge. In the limit of large driving frequency, we find that the distribution of resonant rigidities ~k−5, corresponding to fin shapes with wavenumber k. The input power and output power are proportional to rigidity (for small-to-moderate rigidity) and to pitching frequency (for moderate-to-large frequency). We compare these results with the range of rigidity and flapping frequency for the hawkmoth forewing and the bluegill sunfish pectoral fin.

scholarly journals A vortex sheet based analytical model of the curled wake behind yawed wind turbines

2021 ◽  
Vol 933 ◽  
Author(s):  
Majid Bastankhah ◽  
Carl R. Shapiro ◽  
Sina Shamsoddin ◽  
Dennice F. Gayme ◽  
Charles Meneveau
Keyword(s):  
Analytical Model ◽  
Wind Turbines ◽  
Power Series Expansion ◽  
Wind Farms ◽  
Expansion Method ◽  
Vortex Sheet ◽  
Operating Conditions ◽  
Wake Model ◽  
Ground Effects ◽  
Flow Vortex

Motivated by the need for compact descriptions of the evolution of non-classical wakes behind yawed wind turbines, we develop an analytical model to predict the shape of curled wakes. Interest in such modelling arises due to the potential of wake steering as a strategy for mitigating power reduction and unsteady loading of downstream turbines in wind farms. We first estimate the distribution of the shed vorticity at the wake edge due to both yaw offset and rotating blades. By considering the wake edge as an ideally thin vortex sheet, we describe its evolution in time moving with the flow. Vortex sheet equations are solved using a power series expansion method, and an approximate solution for the wake shape is obtained. The vortex sheet time evolution is then mapped into a spatial evolution by using a convection velocity. Apart from the wake shape, the lateral deflection of the wake including ground effects is modelled. Our results show that there exists a universal solution for the shape of curled wakes if suitable dimensionless variables are employed. For the case of turbulent boundary layer inflow, the decay of vortex sheet circulation due to turbulent diffusion is included. Finally, we modify the Gaussian wake model by incorporating the predicted shape and deflection of the curled wake, so that we can calculate the wake profiles behind yawed turbines. Model predictions are validated against large-eddy simulations and laboratory experiments for turbines with various operating conditions.

Sign in / Sign up

Export Citation Format

Share Document