Fluid Conservation Laws Research Articles

Analogous formulation of electrodynamics and two-dimensional fluid dynamics

AbstractA single, simply stated approximation transforms the equations for a two-dimensional perfect fluid into a form that is closely analogous to Maxwell’s equations in classical electrodynamics. All the fluid conservation laws are retained in some form. Waves in the fluid interact only with vorticity and not with themselves. The vorticity is analogous to electric charge density, and point vortices are the analogues of point charges. The dynamics is equivalent to an action principle in which a set of fields and the locations of the point vortices are varied independently. We recover classical, incompressible, point vortex dynamics as a limiting case. Our full formulation represents the generalization of point vortex dynamics to the case of compressible flow.

On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities

We show how Noether conservation laws can be obtained from the particle relabelling symmetries in the Euler-Poincar\'e theory of ideal fluids with advected quantities. All calculations can be performed without Lagrangian variables, by using the Eulerian vector fields that generate the symmetries, and we identify the time-evolution equation that these vector fields satisfy. When advected quantities (such as advected scalars or densities) are present, there is an additional constraint that the vector fields must leave the advected quantities invariant. We show that if this constraint is satisfied initially then it will be satisfied for all times. We then show how to solve these constraint equations in various examples to obtain evolution equations from the conservation laws. We also discuss some fluid conservation laws in the Euler-Poincar\'e theory that do not arise from Noether symmetries, and explain the relationship between the conservation laws obtained here, and the Kelvin-Noether theorem given in Section 4 of Holm, Marsden and Ratiu, {\it Adv. in Math.}, 1998.

Development of an overset grid computational fluid dynamics solver on graphical processing units

General Purpose computation on Graphics Processing Units (GPGPUs) has gained popularity recently. Graphics Processing Units (GPUs) are being used for computationally intensive and data intensive problems to obtain orders of magnitude speed up in wide range of domains like molecular dynamics, bio-physics, geo-physics and CFD [1–4]. In this paper we discuss the development of a two-dimensional overset grid CFD solver on GPUs for moving body problems and demonstrate orders of magnitude speed-up on single GPU unit as compared to C/FORTRAN solver on a single CPU core. The two-dimensional overset grid CFD solver consists of three modules. A near-body solver module which solves the fluid conservation laws on structured and unstructured mesh systems, an off-body solver module which solves the fluid-conservation laws on an isotropic Cartesian mesh, and the domain connectivity module which manages the interaction between these two mesh systems. The GPU acceleration is extended to all the three modules. We expect this work to be the fore-runner for future development efforts for full three-dimensional Navier–Stokes solutions capable of executing in a heterogeneous parallel environment. Here “heterogeneous parallelism” refers to combination of Message Passing Interface (MPI) based communication for distributed memory systems, and large scale multi-threading using GPUs for shared memory systems.

Conservative unsteady aerodynamic simulation of arbitrary boundary motion using structured and unstructured meshes in time

SUMMARYSimulation of unsteady fluid behaviour with arbitrary boundary motion or topological change remains restricted owing to mesh deformation limitations, and usually requires the application of special techniques using overlapping meshes, sliding planes, remeshing or immersed boundaries. This work presents the application of a spacetime interpretation of the fluid conservation laws that unifies meshes in space and time. This effectively replaces the problem of mesh deformation with the problem of mesh generation and, because connectivity is no longer restricted to being constant in time, any motion or topological change may be simulated. Examples are given applying the method to a piston, a pitching NACA0012 aerofoil, an appearing/disappearing object, a two‐dimensional store separation and a rotation case to validate and then demonstrate the capabilities of the method. Results for the pitching aerofoil case are compared with a conventional moving mesh unsteady computation and shown to be consistent, whereas the demonstration cases show qualitatively correct behaviour and illustrate the general nature of the technique. Copyright © 2011 John Wiley & Sons, Ltd.

Numerical Simulation of Direct Current Glow Discharges for High-Speed Flow Control

Numerical studies of glow discharges were carried out to evaluate their utility for flow control applications. As part of this project, a three-dimensional computer code has been written to solve the fluid conservation laws, the charged particle continuity equations under the drift-diffusion model, and the Poisson equation for the electric potential. Glow discharge calculations have been carried out in the absence of flow, exploring the effects of background pressure, applied magnetic fields, and finite electrodes. With the presence of a Mach 5 crossflow, three-dimensional computations have identified the origin of the downward force in a plasma flap concept as dissipative heating in the vicinity of the cathode, and rough agreement has been obtained between numerical predictions and experimental measurements of the total temperature rise in that region. A simplified actuator model has been applied in the numerical exploration of the control of a Mach 14 compression ramp flow, and suitable actuator locations and power levels have been identified.

On the linearization of systems of conservation laws for fluids at a material contact discontinuity

In order to investigate the linearized stability or instability of compressible flows, as it occurs for instance in Rayleigh–Taylor or Kelvin–Helmholtz instabilities, we consider the linearization at a material discontinuity of a flow modeled by a multidimensional nonlinear hyperbolic system of conservation laws. Restricting ourselves to the plane-symmetric case, the basic solution is thus a one-dimensional contact discontinuity and the normal modes of pertubations are solutions of the resulting linearized hyperbolic system with discontinuous nonconstant coefficients and source terms. While in Eulerian coordinates, the linearized Cauchy problem has no solution in the class of functions, we prove that for a large class of systems of conservation laws written in Lagrangian coordinates and including the Euler and the ideal M.H.D. systems, there exists a unique function solution of the problem that we construct by the method of characteristics.

TVD FINITE‐DIFFERENCE METHODS FOR COMPUTING HIGH‐SPEED THERMAL AND CHEMICAL NON‐EQUILIBRIUM FLOWS WITH STRONG SHOCKS

Partially‐decoupled upwind‐based total‐variation‐diminishing (TVD) finite‐difference schemes for the solution of the conservation laws governing two‐dimensional non‐equilibrium vibrationally relaxing and chemically reacting flows of thermally‐perfect gaseous mixtures are presented. In these methods, a novel partially‐decoupled flux‐difference splitting approach is adopted. The fluid conservation laws and species concentration and vibrational energy equations are decoupled by means of a frozen flow approximation. The resulting partially‐decoupled gas‐dynamic and thermodynamic subsystems are then solved alternately in a lagged manner within a time marching procedure, thereby providing explicit coupling between the two equation sets. Both time‐split semi‐implicit and factored implicit flux‐limited TVD upwind schemes are described. The semi‐implicit formulation is more appropriate for unsteady applications whereas the factored implicit form is useful for obtaining steady‐state solutions. Extensions of Roe's approximate Riemann solvers, giving the eigenvalues and eigenvectors of the fully coupled systems, are used to evaluate the numerical flux functions. Additional modifications to the Riemann solutions are also described which ensure that the approximate solutions are not aphysical. The proposed partially‐decoupled methods are shown to have several computational advantages over chemistry‐split and fully coupled techniques. Furthermore, numerical results for single, complex, and double Mach reflection flows, as well as corner‐expansion and blunt‐body flows, using a five‐species four‐temperature model for air demonstrate the capabilities of the methods.

Helicity conservation laws for fluids and plasmas

Helicity conservation laws in fluid dynamics and magnetohydrodynamics are studied. The rate of fluid helicity dissipation in nonisentropic flow is derived, and the derivation is generalized, introducing the concept of chirality and reinterpreting the Ertel-Katz law as one of a family of laws dictating the variation of the rate of dissipation of some helicity along streamlines. Woltjer's two helicity conservation laws in MHD are reviewed, and the covariant derivation of the second is given. The two Oron helicities are introduced, their conserved character is established, and a type B law arising from the second Oron helicity's conservation is proved. Departures from perfect MHD are addressed, estimating the magnetic helicity (MH) dissipation in magnetized fluids with finite conductivity and deriving a type-B law for the MH dissipation rate in a particularly simple kind of imperfect MHD flow. An Ertel-Katz type law is derived for poorly conducting magnetized fluids.

Anomalous transport due to long-lived fluctuations in plasma Part 1. A general formalism for two-time fluctuations

A general formalism for describing two-time fluctuations in magnetized plasma is presented. Two-time expectations of one-body operators (phase functions) are written in terms of the phase space density autocorrelation functionwhere δN is the fluctuation in the singular Klimontovich microdensity. It is shown that is the first member of a set of two-time quantitieswhich collectively obeys the linearized BBGKY cumulant hierarchy in the (Xi, t) variables, with initial conditions successively smaller in the plasma parameter . We study in detail the case of fluctuations in thermal equilibrium, although the general formalism holds also for the non-equilibrium case. To lowest order in εP, Γ obeys the linearized Vlasov equation. From this are recovered all of Rostoker's results for fluctuations excited by Cherenkov emission and absorbed by Landau damping, as well as a constructive proof of the test particle superposition principle. To first order, Γ obeys (in the Markovian approximation) the linearized Balescu-Guernsey-Lenard equation. For frequencies and wavenumbers in the hydrodynamic regime, the velocity moments of Γ obey linearized fluid equations with classical transport coefficients (i.e. essentially those computed by Braginskii in the 3-D case). It has been found that the classical theory is in disagreement with certain computer and laboratory experiments performed in strong magnetic fields. This defect is attributed to the absence in the classical theory of contributions to the collision operator, hence transport coefficients, of fluctuations long-lived on the Vlasov scale. Analogous difficulties arise in the theory of hydrodynamics in neutral fluids. To improve the plasma theory, a renormalization of the two-time hierarchy is proposed which sums selected terms from all orders in εP and thus treats the hydrodynamic fluctuations self-consistently. The resulting theory retains appropriate fluid conservation laws, thereby avoiding erroneous results encountered in certain diffusing orbit theories, when the fluid viscosity is indiscriminantly replaced by the test particle diffusion coefficient. In order to explain the results of the computer simulations, the theory is applied in part 2 to the problem of anomalous hydrodynamic contributions to the transport coefficients.