Navier-Stokes Equations For Compressible Fluids Research Articles

Application of the Hierarchical Functions Expansion Method for the Solution of the Two Dimensional Navier-Stokes Equations for Compressible Fluids in High Velocity

This work presents a new application for the Hierarchical Function Expansion Method for the solution of the Navier-Stokes equations for compressible fluids in two dimensions and in high velocity. This method is based on the finite elements method using the Petrov-Galerkin formulation, know as SUPG (Streamline Upwind Petrov-Galerkin), applied with the expansion of the variables into hierarchical functions. To test and validate the numerical method proposed as well as the computational program developed simulations are performed for some cases whose theoretical solutions are known. These cases are the following: continuity test, stability and convergence test, temperature step problem, and several oblique shocks. The objective of the last cases is basically to verify the capture of the shock wave by the method developed. The results obtained in the simulations with the proposed method were good both qualitatively and quantitatively when compared with the theoretical solutions. This allows concluding that the objectives of this work are reached.

On global behavior of weak solutions to the Navier-Stokes equations of compressible fluid for γ = 5 / 3 $\gamma=5/3$

In this article, we consider the global behavior of weak solutions of the Navier-Stokes equations of a compressible fluid in a bounded domain driven by bounded forces for the adiabatic constant $\gamma=5/3$ . Under the condition of a small mass depending on the given forces, we prove the existence of bounded absorbing sets of weak solutions, and thus we further get global bounded trajectories and global attractors to the weak solutions.

Anomalous scaling of passive scalar fields advected by the Navier-Stokes velocity ensemble: effects of strong compressibility and large-scale anisotropy.

The field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities (the density and the tracer fields) advected by a random turbulent velocity field. The latter is governed by the Navier-Stokes equation for compressible fluid, subject to external random force with the covariance ∝δ(t-t')k(4-d-y), where d is the dimension of space and y is an arbitrary exponent. The original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models; the corresponding renormalization group equations possess infrared attractive fixed points. It is shown that various correlation functions of the scalar field, its powers and gradients, demonstrate anomalous scaling behavior in the inertial-convective range already for small values of y. The corresponding anomalous exponents, identified with scaling (critical) dimensions of certain composite fields ("operators" in the quantum-field terminology), can be systematically calculated as series in y. The practical calculation is performed in the leading one-loop approximation, including exponents in anisotropic contributions. It should be emphasized that, in contrast to Gaussian ensembles with finite correlation time, the model and the perturbation theory presented here are manifestly Galilean covariant. The validity of the one-loop approximation and comparison with Gaussian models are briefly discussed.

Numerical Simulation of Cavitating Water Jet by a Compressible Mixture Flow Method

Concerned on the numerical simulation of high-speed water jet accompanied with intensive cavitation, a practical compressible mixture flow method is developed by coupling a simplified estimation of bubble cavitation and a flexible compressible flow computation procedure. Two-phase fluid media of cavitating flow are treated as a mixture of liquid and bubbles, and the mean flow of the mixture is computed by applying Reynolds Averaged Navier-Stokes equations for compressible fluids considering the effect of bubble expansion a/o contraction. The intensity of cavitation is evaluated by gas volume fraction which is governed by the compressibility of bubble-liquid mixture at the current status of mean flow field. High-speed submerged water jets issuing from a convergent-divergent nozzle are treated under different cavitation conditions. The results demonstrate that pressure decreases in the convergent section and cavitation occurs initially in the low-pressure region behind the throat near the wall. The gas volume fraction of cavitation bubbles reaches to 0.5 locally when the cavitation number is decreased to 0.1, and cavitation bubbles generated in the shear layer near the throat flow downstream along the jet periphery. Under the effect of cavitation bubbles the mass discharge coefficient of water jet decreases obviously compared to the case of no-cavitating one although the maximum velocity at the throat increases.

Large eddy simulation of a shocked gas cylinder instability induced turbulence

The Navier-Stokes equations for compressible fluid are solved with the operator splitting technique and LES (large eddy simulation) with the Smagorinsky model. A computational code MVFT (multi-viscosity-fluid and turbulence) is developed to study hydrodynamic instability and the induced turbulent mixing for multi compressible fluid. In order to validate the code MVFT, the LANL’s shock tube experiment of shocked SF6 gas cylinder is simulated with the initial state of SF6 gas cylinder described by dissipative ITL (interface transition layer). It is shown that the width and height of gas cylinder calculated with MVFT are closer to the experimental results than RAGE, and that the velocities of upstream edge, downstream edge and vortex edge agree with the experimental results, and are appreciably smaller than the RAGE results. The code MVFT has been preliminarily validated.

On green's function for hyperbolic-parabolic systems

We study the Green's function for a general hyperbolic-parabolic system, including the Navier-Stokes equations for compressible fluids and the equations for magneto hydrodynamics. More generally, we consider general systems under the basic Kawashima Shizuta type of conditions. The first result is to make precise the secondary waves with subscale structure, revealing the nature of coupling of waves pertaining to different characteristic families. The second result is on the continuous differentiability of the Green's function with respect to a small parameter when the coefficients of the system are smooth functions of that parameter. The results significantly improve previous results obtained by the authors.

G1301 気泡混合流モデルによるベンチューリノズルにおけるキャビテーション流れの数値解析(2)(GS13 数値シミュレーション,一般セッション)

A compressible bubble-liquid mixture flow model is developed for computing high-speed cavitation flows by coupling a bubble dynamic calculation based on the Two-Fluids Three-Pressure model and a compressible mixture flow procedure. General Navier-Stokes equations for compressible fluid are employed to simulate the flow of two-phase mixture. State equations of liquid and gas phases are adopted and a relation for the mixture density and pressure is derived. As an example, cavitating and no-cavitating flows in a Venturi nozzle are treated and the capability of present model is demonstrated.

Detection of multiple solutions using a mid-cell back substitution technique applied to computational fluid dynamics

Computational models for fluid flow based on the Navier-Stokes equations for compressible fluids led to numerical procedures requiring the solution of simultaneous non-linear algebraic equations. These give rise to the possibility of multiple solutions, and hence there is a need to monitor convergence towards a physically meaningful flow field. The number of possible solutions that may arise is examined, and a mid-cell back substitution technique (MCBST) is developed to detect and avoid convergence towards apparently spurious solutions. The MCBST was used successfully for flow modelling in micron-sized flow passages, and was found to be particularly useful in the early stages of computation, optimizing the speed of convergence.

A SURVEY OF THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

This paper presents mathematical properties of solutions to the Navier-Stokes equations for compressible fluids. We first review existence results for the Cauchy problem, and describe some regularity properties of solutions in the presence of possibly vanishing densities. Finally, we address the problem of the low Mach number limit leading to incompressible models.

Hamiltonian and onsageristic approaches in the nonlinear theory of fluid-permeable elastic continua

A new system of equations governing nonlinear dynamics of fluid-permeable poroelastic media is derived on the basis of Hamilton's principle for reversible effects and the Onsager-Sedov approach for irreversible effects. The equations are in Eulerian variables, suitable for dealing with fluid and solid phenomena simultaneously. The classical (Murnaghan-like) equations of nonlinear elasticity, as well as the governing equations of the ideal and Navier-Stokes fluids are shown to be special cases of the general governing system. It is well-known that the Navier-Stokes equations of compressible fluid provide a correct self-consistent basis for studying a wide variety of nonlinear effects in fluids. The authors believe that the governing system proposed here provides the same opportunities for various nonlinear effects in poro-elastic fluid-penetrable media.

Adaptive methods on unstructured grids for Euler and Navier-Stokes equations

Adaptive methods on unsaturated, triangulated grids are considered for solution of the 2-D Euler and Navier-Stokes equations of compressible fluids. The finite-volume method is used for conservative discretization. The equations are approximated by central or upwind discretizations and solved with an explicit Runge-Kutta scheme. Different generation techniques are considered for the construction of suitable initial meshes. The main focus is on the study of adaptive grid methods. corresponding to the local requirements of inviscid and viscous solutions. Central grid refinement and adaptive cell-orientation have been shown to be well-suited for computation of inviscid flows. More sophisticated adaptation concepts are necessary for solution of the Navier-Stokes equations, due to the different scale lengths in viscous flows. For such solutions a method based on virtual stretching is presented, which enables anisotropic grid according to the resolution requirements of different criteria. The capability of these methods is demonstrated by a number of computed flow results.

Global weak solvability of a regularized system of the Navier-Stokes equations for compressible fluid

The paper contains the proof of global existence of weak solutions to the mixed initial-boundary value problem for a certain modification of a system of equations of motion of viscous compressible fluid. The modification is based on an application of an operator of regularization to some terms appearing in the system of equations and it does not contradict the laws of fluid mechanics. It is assumed that pressure is a known function of density. The method of discretization in time is used and finally, a so called energy inequality is derived. The inequality is independent on the regularization used.

Navier-stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case

We consider the equations which describe the motion of a viscous compressible fluid, taking into consideration the case of inflow and/or outflow through the boundary. By means of some a priori estimates we prove the existence of a global (in time) solution. Moreover, as a consequence of a stability result, we show that there exist a periodic solution and a stationary solution.

Three-dimensional calculation of transonic viscous flows by an implicit method

A new implicit finitevolume method for solving the three-dimensional Navier-Stokes equations for compressible fluids is presented and applied to the calculation of flows over swept wings at moderate Reynolds numbers. The method makes use of an explicit predictor-corrector scheme to calculate a provisional value, which is then corrected by using an implicit operator. The explicit part is a generalization of the centered Thornmen scheme, while the implicit operator is a combination of the inviscid schemes proposed by Lerate and the viscous treatment of Beam and Warming. Results are here presented for a swept wing between two parallel walls and for the finite wing ONERA M6 at Mx =0.8, zero incidence and Re^ = 1000. The computations are performed with a time-step about 5 to 10 times greater than the explicit step.

Numerical simulation of a compressible turbulent boundary layer over a conductive wall with line heat source

Abstract A conjugated problem of supersonic turbulent flow over a conductive solid wall with an embedded line heat source has been investigated as a model of a separation detector and skin friction gage. The 2-D Navier-Stokes equations for compressible fluid, including a two layer eddy viscosity model, are solved simultaneously with the heat transfer equation for the solid, written in general coordinates. The effect of the interface boundary condition on the stability of the implicit scheme of the flow field has been checked. A careful investigation of the effect of heat source strength, solid and fluid conductivity and Mach and Reynolds numbers on flow and temperature fields has been performed. The results of this investigation may be used to design an optimal gage with a minimum influence on the flow field.