Equations In Primitive Variables Research Articles

Port-Hamiltonian formulations of the incompressible Euler equations with a free surface

In this paper, we present port-Hamiltonian formulations of the incompressible Euler equations with a free surface governed by surface tension and gravity forces, modelling e.g. capillary and gravity waves and the evolution of droplets in air. Three sets of variables are considered, namely (v,Σ), (η,ϕ∂,Σ) and (ω,ϕ∂,Σ), with v the velocity, η the solenoidal velocity, ϕ∂ a potential, ω the vorticity, and Σ the free surface, resulting in the incompressible Euler equations in primitive variables and the vorticity equation. First, the Hamiltonian formulation for the incompressible Euler equations in a domain with a free surface combined with a fixed boundary surface with a homogeneous boundary condition will be derived in the proper Sobolev spaces of differential forms. Next, these results will be extended to port-Hamiltonian formulations allowing inhomogeneous boundary conditions and a non-zero energy flow through the boundaries. Our main results are the construction and proof of Dirac structures in suitable Sobolev spaces of differential forms for each variable set, which provides the core of any port-Hamiltonian formulation. Finally, it is proven that the state dependent Dirac structures are related to Poisson brackets that are linear, skew-symmetric and satisfy the Jacobi identity.

Well-conditioned ultraspherical and spectral integration methods for resolvent analysis of channel flows of Newtonian and viscoelastic fluids

Modal and nonmodal analyses of fluid flows provide fundamental insight into the early stages of transition to turbulence. Eigenvalues of the dynamical generator govern temporal growth or decay of individual modes, while singular values of the frequency response operator quantify the amplification of disturbances for linearly stable flows. In this paper, we develop well-conditioned ultraspherical and spectral integration methods for frequency response analysis of channel flows of Newtonian and viscoelastic fluids. Even if a discretization method is well-conditioned, we demonstrate that calculations can be erroneous if singular values are computed as the eigenvalues of a cascade connection of the frequency response operator and its adjoint. To address this issue, we utilize a feedback interconnection of the frequency response operator with its adjoint to avoid computation of inverses and facilitate robust singular value decomposition. Specifically, in contrast to conventional spectral collocation methods, the proposed method (i) produces reliable results in channel flows of viscoelastic fluids at high Weissenberg numbers (∼500); and (ii) does not require a staggered grid for the equations in primitive variables.

An Energy Consistent Discretization of the Nonhydrostatic Equations in Primitive Variables

AbstractWe derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal discretization. The formulation is a version of Dubos and Tort (2014, https://doi.org/10.1175/MWR-D-14-00069.1) rewritten in terms of primitive variables. It is valid for terrain following mass or height coordinates and for both Eulerian or vertically Lagrangian discretizations. The discretization relies on an extension to Simmons and Burridge (1981, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2) vertical differencing, which we show obeys a discrete derivative product rule. This product rule allows us to simplify the treatment of the vertical transport terms. Energy conservation is obtained via a term‐by‐term balance in the kinetic, internal, and potential energy budgets, ensuring an energy‐consistent discretization up to time truncation error with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a horizontal explicit vertically implicit implicit‐explicit time stepping algorithm.

Optimum Fin Parameters of Radial Heat Sinks Subjected to Natural Convection

Free convection from an upward facing radial heat sink with fins at an equal angular gap attached to an isothermal base has been investigated numerically. The governing equations in primitive variables were changed to vorticity-vector potential formulation, and an in-house code was developed using finite difference technique. To close the computational domain, two pseudo boundaries were considered. Length, height, and number of fins strongly influence the rate of heat transfer while the fin thickness has a marginal role. As the fin length increases, the rate of heat transfer first increases and then remains almost unaffected. However, the active length of the fins depends on the strength of buoyancy. Heat transfer continuously increases with fin height but with diminishing effect. Adding more number of fins has two opposing effects. It provides more surface area for convection, but at the same time, the induced air is unable to reach the interior of the heat sink making the inner portion of the fins inoperative. As a result of these two opposing influences, heat transfer increases in the beginning and then decreases as more fins are added. This article suggests various fin parameters to achieve maximum cooling. In addition, one can estimate the rate of cooling to be achieved by any radial heat sink.

Efficient discretizations for the EMAC formulation of the incompressible Navier–Stokes equations

We study discretizations of the incompressible Navier–Stokes equations, written in the newly developed energy–momentum–angular momentum conserving (EMAC) formulation. We consider linearizations of the problem, which at each time step will reduce the computational cost, but can alter the conservation properties. We show that a skew-symmetrized linearization delivers the correct balance of (only) energy and that the Newton linearization conserves momentum and angular momentum, but conserves energy only up to the nonlinear residual. Numerical tests show that linearizing with 2 Newton steps at each time step is very effective at preserving all conservation laws at once, and giving accurate answers on long time intervals. The tests also show that the skew-symmetrized linearization is significantly less accurate. The tests also show that the Newton linearization of EMAC finite element formulation compares favorably to other traditionally used finite element formulation of the incompressible Navier–Stokes equations in primitive variables.

Buoyancy-induced flow and heat transfer in multilayered cavities with openings

PurposeThe purpose of this study is to investigate the flow interaction between the cavities and its impact on heat transfer. The role of the openings is examined and three strategies are considered: one opening, two openings on single side and two openings on double sides.Design/methodology/approachA two-dimensional laminar natural convection heat transfer in multilayered open cavities was numerically investigated. The governing equations in primitive variables were discretized by the finite volume method and solved by SIMPLE algorithm.FindingsThe results show that for the cavities with one opening, the flow in the cavities is connected with each other. The exhaust hot fluid from the lower cavity was entrained into the upper cavities by thermal buoyancy and hence the heat transfer in the upper cavities was decreased because of thermal accumulation. Two openings on the single side could strengthen the flow interaction between the cavities and then enhance the heat transfer. However, the double-sided openings eliminated the flow interaction between the cavities and thus the fluid flow and heat transfer characteristics in all cavities are independent. It was concluded that the flow interaction between the multilayered open cavities has an importance effect on the heat transfer in the cavities.Originality/valueThe flow interaction between the multilayered open cavities was illustrated. The effect of flow interaction on the heat transfer in the cavities was investigated. The role of openings in the flow interaction and heat transfer in cavities was explored. The cavity below affects above cavity for the openings on single side. No interaction exists between the cavities with openings on double sides.

Algebraic splitting methods for the steady incompressible Navier–Stokes equations at moderate Reynolds numbers

The unsteady incompressible Navier–Stokes equations in primitive variables are often numerically solved by segregating the computation of velocity and pressure, according to either functional analysis arguments following the pioneering work of A.J. Chorin and R. Temam (STD, Split-Then-Discretize paradigm) or linear algebra arguments based on the inexact block factorization of the discretized problem (DTS, Discretize-Then-Split paradigm). The presence of the time derivative allows for the calibration of an appropriate approximation of the pseudo-differential operator of the pressure problem and excellent results in terms of both accuracy and efficiency have been obtained as witnessed by the abundant literature. The extension of the same segregated approach to the steady Navier–Stokes equations is unclear, unless a pseudo-time advancing formulation is undertaken. In this paper we present a methodology for a segregated computation of the primitive variables in a genuinely steady formulation, so to avoid iterations to get to the steady limit. The approach is largely inspired by the algebraic factorization of the unsteady problem (DTS approach), yet we detail specific settings required by the absence of the velocity time-derivative. The basic idea relies on the introduction of some parameters in a modified Picard linearization. We discuss stability bounds and the convergence of the segregated method to the unsplit solution. Several numerical results on different test cases confirm the efficiency of the procedure.

Numerical simulation of thermomagnetic convection of cold water near its density maximum in a square enclosure under a magnetic quadrupole field

Natural convection of cold water near its density maximum in a square enclosure is studied numerically under the influence of a magnetic quadrupole field without gravity. A generalized model which includes a non-Boussinesq parabolic density–temperature relationship is established. The governing equations in primitive variables are discretized using the finite-volume method and solved using the SIMPLE algorithm. The effects of magnetic force number, Rayleigh number and density inversion parameter on flow and heat transfer characteristics are analyzed. The results show that the primary flow pattern depends mainly on the density inversion parameter. Multi-cellular flow structures are observed for certain ranges of density inversion parameter independent of the value of Rayleigh number and magnetic force number. The heat transfer changes non-monotonically under combined actions of the quadrupole magnetic field and density inversion.

Recent Results from Analysis of Flow Structures and Energy Modes Induced by Viscous Wave around a Surface-Piercing Cylinder

Due to its relevance in ocean engineering, the subject of the flow field generated by water waves around a vertical circular cylinder piercing the free surface has recently started to be considered by several research groups. In particular, we studied this problem starting from the velocity-potential framework, then the implementation of the numerical solution of the Euler equations in their velocity-pressure formulation, and finally the performance of the integration of the Navier-Stokes equations in primitive variables. We also developed and applied methods of extraction of the flow coherent structures and most energetic modes. In this work, we present some new results of our research directed, in particular, toward the clarification of the main nonintuitive character of the phenomenon of interaction between a wave and a surface-piercing cylinder, namely, the fact that the wave exerts its maximum force and exhibits its maximum run-up on the cylindrical obstacle at different instants. The understanding of this phenomenon becomes of crucial importance in the perspective of governing the entity of the wave run-up on the obstacle by means of wave-flow-control techniques.

Convergence of the Marker-and-Cell Scheme for the Incompressible Navier–Stokes Equations on Non-uniform Grids

We prove in this paper the convergence of the Marker and cell (MAC) scheme for the dis-cretization of the steady-state and unsteady-state incompressible Navier-Stokes equations in primitive variables on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven ; they yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step of which tends to zero. We then establish that the limit is a weak solution to the continuous problem.

The intermittent nature of the laminar separation bubble on a cylinder in uniform flow

Abstract Flow past a circular cylinder in a uniform flow is investigated for 1 × 10 4 ≤ Re ≤ 4 × 10 5 . A stabilized finite element method is used to solve the incompressible flow equations in primitive variables in three dimensions. The computations capture the phenomenon of drag crisis: a significant reduction in drag with increase in Re in the critical regime. The mechanism for this decrease in drag during the drag crisis is explored. It is found that the transition of the boundary layer from a laminar to turbulent state, as well as the formation of the laminar separation bubble (LSB), is intermittent. The LSB does not exist in the sub-critical regime while it appears at all times beyond the critical regime. The frequency of its appearance as well as the duration of its stay, in the critical Re regime, increases with increase in Re . This is established by studying the rms of the high pass filtered fluctuations in the surface pressure and flow close to the cylinder. A procedure to capture these fluctuations due to the shear layer activity responsible for the formation of LSB is proposed and implemented. It is utilized to estimate the intermittency factor of the LSB at various Re . It is found that the intermittency can be utilized to explain the variation of mean drag with Re . Weakening of vortex shedding is observed in critical regime and beyond.

Transient Analysis of Coupled Natural Convection and Surface Thermal Radiation in a Tilted Open Cavity

Numeric results of the transient heat transfer by natural convection and surface thermal radiation in a tilted open cavity are presented. The conservation equations in primitive variables are solved using the finite volume method and the SIMPLEC algorithm. The transient results are obtained for a Rayleigh number of 106 and five inclination angles (0°, 45°, 90°, 120° and 180°. The numerical model predicts flow instabilities for inclination angle of 0°, which avoids reaching the steady state. The steady state is reached after a long time for ϕ=180°.

Localized method of approximate particular solutions for solving unsteady Navier–Stokes problem

The localized method of approximate particular solutions (LMAPS) is proposed to solve two-dimensional transient incompressible Navier–Stokes systems of equations in primitive variables. The equations contain the Laplacian operator. In avoiding ill-conditioning problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear system within local supporting domain in which five nearest neighboring points and multiquadrics are used for interpolation. Then local matrices are reformulated in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The method is assessed on driven cavity problem and flow around cylinder. The numerical experiments show that the newly developed LMAPS is suitable for solving incompressible Navier–Stokes equations with high accuracy and efficiency.

Numerical Simulations of Wave-Induced Flow Fields around Large-Diameter Surface-Piercing Vertical Circular Cylinder

A computational analysis is performed on the diffraction of water waves induced by large-diameter, surface-piercing, vertical circular cylinder. With reference to linear-wave cases, the phenomenon is preliminarly considered in terms of velocity potential, a simplified theoretical framework in which both hypotheses of inviscid fluid and irrotational flow are incorporated. Then, and as a first-approximation analysis, the Euler equations in primitive variables are considered (a framework in which the fluid is still handled as inviscid, but the field can be rotational). Finally, the real-fluid behavior is analyzed, by numerically integrating the full Navier-Stokes equations (viscous fluid and rotational field) in their velocity-pressure formulation, by following the approach of the Direct Numerical Simulation (DNS, no models are used for the fluctuating portion of the velocity field). For further investigation of the flow fields, the swirling-strength criterion for flow-structure extraction, and the Karhunen-Loève (KL) decomposition technique for the extraction of the most energetic flow modes respectively, are applied to the computed fields. It is found that remarkable differences exist between the wave-induced fields, as derived within the different computing frameworks tested.

The Field of Flow Structures Generated by a Wave of Viscous Fluid Around Vertical Circular Cylinder Piercing the Free Surface

The diffraction of water waves induced by a large-diameter, surface-piercing, vertical circular cylinder is studied numerically. The Navier-Stokes equations in primitive variables are considered for the simulation of a given wave case, and the technique is followed of the Direct Numerical Simulation (DNS). The criterion of the imaginary part of the complex-eigenvalue pair of the velocity-gradient tensor for the extraction of the flow vortical structures is applied to the computed fields, so unveiling a complex configuration of structures at the free surface, and at the cylinder external walls. The most energetic modes of the flow are further extracted from the DNS-simulated fields by using the Karhunen-Loève decomposition (KL). A “reduced” velocity field is reconstructed using the first three most energetic eigenfunctions of the decomposition, and its evolution is followed through a sequence of time steps.

Effect of Coil Inclination on Thermomagnetic Convection of Air in a Porous Cubic Enclosure Under Microgravity Environment

The thermomagnetic convection of air in a porous cubic enclosure with an electric coil inclined in general orientations was numerically investigated under microgravity environment. Biot-Savart’s was used to calculate magnetic field. The governing equations in primitive variables were discretized by the finite-volume method and solved by the SIMPLE algorithm. The Darcy model was used to solve the momentum equations. The results show that both the magnetic force and the coil inclination have significant effect on the flow field and heat transfer in a porous cubic enclosure; the natural convection heat transfer of air can be enhanced or controlled by applying gradient magnetic field

Theoretical and experimental study of natural convection with surface thermal radiation in a side open cavity

In this work a theoretical and experimental study of heat transfer by natural convection and thermal radiation on a solar open cubic cavity-type receiver is presented. The theoretical study consists on solving the laminar natural convection and the surface thermal radiation on a square open cavity at one end. The overall continuity, momentum, and energy equations in primitive variables are solved numerically by using the finite-volume method and the SIMPLEC algorithm. The thermophysical properties of the fluid are considered, for the first case, as temperature dependent in all the governing equations, and for the second case, constant, except for the density at the buoyancy term (Boussinesq approximation), with the purpose of comparing the results of both theoretical models with experimentally obtained results. Numerical calculations are conducted for Rayleigh number (Ra) values in the range of 104–106. The temperature difference between the hot wall and the bulk fluid (ΔT) is varied between 10 and 400 K, and is represented as a dimensionless temperature difference (φ) for the purpose of generalization of the trends observed. Experimental results include air temperature measurements inside the receiver. These results are compared with theoretically obtained air temperatures, and the average deviation between both results is around 3.0%, when using the model with variable thermophysical properties, and is around 5.4% when using the Boussinesq approximation.

Conservation properties and potential systems of vorticity-type equations

Partial differential equations of the form \documentclass[12pt]{minimal}\begin{document}$\mathop {\hbox{\rm div}}{{\bm N}} =0$\end{document}divN=0, \documentclass[12pt]{minimal}\begin{document}${{\bm N}}_t + \mathop {\hbox{\rm curl}}{{\bm M}}=0$\end{document}Nt+curlM=0 involving two vector functions in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3 depending on t, x, y, z appear in different physical contexts, including the vorticity formulation of fluid dynamics, magnetohydrodynamics (MHD) equations, and Maxwell's equations. It is shown that these equations possess an infinite family of local divergence-type conservation laws involving arbitrary functions of space and time. Moreover, it is demonstrated that the equations of interest have a rather special structure of a lower-degree (degree two) conservation law in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^4(t,x,y,z)$\end{document}R4(t,x,y,z). The corresponding potential system has a clear physical meaning. For the Maxwell's equations, it gives rise to the scalar electric and the vector magnetic potentials; for the vorticity equations of fluid dynamics, the potentialization inverts the curl operator to yield the fluid dynamics equations in primitive variables; for MHD equations, the potential equations yield a generalization of the Galas-Bogoyavlenskij potential that describes magnetic surfaces of ideal MHD equilibria. The lower-degree conservation law is further shown to yield curl-type conservation laws and determined potential equations in certain lower-dimensional settings. Examples of new nonlocal conservation laws, including an infinite family of nonlocal material conservation laws of ideal time-dependent MHD equations in 2+1 dimensions, are presented.

Magnetic and Gravitational Convection of Air in a Porous Cubic Enclosure with a Coil Inclined Around the Y Axis

The natural convection heat transfer of air in a porous media can be controlled by gradient magnetic field. Thermomagnetic convection of air in a porous cubic enclosure with an electric coil inclined around the \(Y\) axis was numerically investigated. The Biot–Savart law was used to calculate the magnetic field. The governing equations in primitive variables were discretized by the finite-volume method and solved by the SIMPLE algorithm. The flow and temperature fields for the air natural convection were presented and the mean Nusselt number on the hot wall was calculated and compared. The results show that both the magnetic force and coil inclination have significant effect on the flow field and heat transfer in a porous cubic enclosure, the natural convection heat transfer of air can be enhanced or controlled by applying gradient magnetic field.

Numerical solution of the incompressible Navier–Stokes equations in primitive variables and velocity–vorticity formulation

A finite-difference method is presented for the numerical solution of the Navier–Stokes equations of motion of a viscous incompressible fluid in two dimensions in primitive-variables and velocity–vorticity formulation. For the case of primitive-variables, using a staggered grid and introducing an auxiliary function of the coordinate system and considering the form of the initial equation on lines passing through the nodal point (x0, y0) and parallel to the coordinate axes, we can separate it into two parts that are finally reduced to ordinary linear differential equations, one for each dimension. Discretization of these equations leads to a system of linear equations in n-unknowns which is solved by an iterative technique and the method converges rapidly giving satisfactory results. For the pressure variable we consider a pressure Poisson equation with suitable Neumann type boundary conditions. In case of velocity–vorticity formulation similar procedure is used, for a collocated grid. Numerical results up to Reynolds number 5000, confirming the accuracy of the proposed method, are presented for configurations of interest, such as Poiseuille flow, lid-driven cavity flow in square domain and flow in a backward-facing step in the presence of a pressure gradient.