A thermodynamically consistent model for cavitating flows of compressible fluids

This work presents a logically consistent thermodynamic model to describe the isothermal cavitation phenomenon in compressible fluid flows. The fluid is regarded as a continuum mixture of liquid and vapor phases (both having the same velocity and temperature), which can or cannot coexist at a same material point and time. The volume fraction is considered as an internal variable and its constraint is treated as a material property, being part of the constitutive relations. Dissipative effects associated with the liquid-vapor phase change transformation and with the vapor volume fraction evolution are taken into account in such a way the Second Law of Thermodynamics is always satisfied. It is shown that the dissipative mechanisms are responsible for a cavitation threshold rule that leads to cavity formation under completely different situations from that characterized by the traditional (reversible) theory. The potentiality of the model as well as its basic features are illustrated and highlighted through a simple numerical example. It is demonstrated that the irreversibility associated with the phase change transformation may be seen as an intermediate case between two physically different non-dissipative situations. One in which the phase change takes place at a constant pressure (the saturated vapor pressure) and the other in which the vapor expands and contracts in the mixture without transforming into liquid.     

A numerical method to simulate turbulent cavitating flows

The objective of this paper is to develop a numerical method for simulating multiphase cavitating flows on unstructured grids. The multiphase medium is represented using a homogeneous mixture model that assumes thermal equilibrium between the liquid and vapor phases. We develop a predictor–corrector approach to solve the governing Navier–Stokes equations for the liquid/vapor mixture, together with the transport equation for the vapor mass fraction. While a non-dissipative and symmetric scheme is used in the predictor step, a novel characteristic-based filtering scheme with a second order TVD filter is developed for the corrector step to handle shocks and material discontinuities in non-ideal gases and mixtures. Additionally, a sensor based on vapor volume fraction is proposed to localize dissipation to the vicinity of discontinuities. The scheme is first validated for simple one dimensional canonical problems to verify its accuracy in predicting jump conditions across material discontinuities and shocks. It is then applied to two turbulent cavitating flow problems – over a hydrofoil using RANS and over a wedge using LES. Our results show that the simulations are in good agreement with experimental data for the above tested cases, and that the scheme can be successfully applied to both RANS and LES methodologies.     

Shock wave interactions with particles and liquid fuel droplets

Shock waves traveling through a multiphase flow environment are studied numerically using the Flux Corrected Transport (FCT) algorithm. Both solid particles and liquid droplets are used as the dispersed phase with their trajectories being computed using a Lagrangian tracking scheme. The phases are coupled by including source terms which account for mass transfer, momentum, and energy exchange from the dispersed phase in the governing equations of motion for the gas phase. For solid particles, droplet size effects are examined at constant mass loading. Deceleration of the shock wave is observed with effects increasing with decreasing particle size. The equilibrium velocity attained is found to agree with analytical results for an equivalent dense gas with a modified specific heat ratio. For liquid droplets, a droplet breakup model is introduced and the results show a faster attenuation rate than with the solid particle model. The inclusion of vaporization to the breakup model is seen to increase the attenuation rate but does not alter the final equilibrium velocity. When an energy release model is used in the simulations, behavior resembling a detonation is observed under certain conditions, with energy release coupling with and accelerating the shock front. Received 17 July 2000 / Accepted 20 August 2002 / Published online 4 December 2002 Correspondence to: Dr. K. Kailasanath (e-mail: kailas@lcp.nrl.navy.mil)     

Direct numerical simulation of compressible turbulent flows

This paper reviews the authors＇ recent studies on compressible turbulence by using direct numerical simulation （DNS）,including DNS of isotropic（decaying） turbulence, turbulent mixing-layer,turbulent boundary-layer and shock/boundary-layer interaction.Turbulence statistics, compressibility effects,turbulent kinetic energy budget and coherent structures are studied based on the DNS data.The mechanism of sound source in turbulent flows is also analyzed. It shows that DNS is a powerful tool for the mechanistic study of compressible turbulence.     

Multiphase fluid dynamics and transport processes of low capillary number cavitating flows

To better understand the multiphase fluid dynamics and associated transport processes of cavitating flows at the capillary number of 0.74 and 0.54, and to validate the numerical results, a combined computational and experimental investigation of flows around a hydrofoil is studied based on flow visualizations and time-resolved interface movement. The computational model is based on a modified RNG k-ε model as turbulence closure, along with a vapor-liquid mass transfer model for treating the cavitation process. Overall, favorable agreement between the numerical and experimental results is observed. It is shown that the cavi- tation structure depends on the interaction of the water-vapor mixture and the vapor among the whole cavitation stage, the interface between the vapor and the two-phase mixture exhibits substantial unsteadiness. And, the adverse motion of the interface relates to pressure and velocity fluctuations inside the cavity. In particular, the velocity in the vapor region is lower than that in the two-phase region.     

Assessment of WENO-extended two-fluid modelling in compressible multiphase flows

The two-fluid modelling based on an advection-upwind-splitting-method (AUSM)-family numerical flux function, AUSM⁺-up, following the work by Chang and Liou [Journal of Computational Physics 2007;225: 840–873], has been successfully extended to the fifth order by weighted-essentially-non-oscillatory (WENO) schemes. Then its performance is surveyed in several numerical tests. The results showed a desired performance in one-dimensional benchmark test problems: Without relying upon an anti-diffusion device, the higher-order two-fluid method captures the phase interface within a fewer grid points than the conventional second-order method, as well as a rarefaction wave and a very weak shock. At a high pressure ratio (e.g. 1,000), the interpolated variables appeared to affect the performance: the conservative-variable-based characteristic-wise WENO interpolation showed less sharper but more robust representations of the shocks and expansions than the primitive-variable-based counterpart did. In two-dimensional shock/droplet test case, however, only the primitive-variable-based WENO with a huge void fraction realised a stable computation.     

Quasi-static cylindrical cavity expansion in an elastoplastic compressible Mises solid

The self-similar elastoplastic field induced by quasi-static expansion of a pressurized cylindrical cavity is investigated for Mises solids under the assumption of plane-strain. Material behavior is modeled by the elastoplastic J₂ flow theory with the standard hypoelastic version. The theory accounts for elastic-compressibility and allows for arbitrary strain-hardening (or softening) in the plastic range. A formulation of the exact governing equations is presented and analyzed in detail for the remote elastic field and for asymptotic plastic behavior near the cavity wall, along with numerical investigations for the entire deformation zone. An analytical solution was obtained under the axially-hydrostatic assumption (axial stress coincides with hydrostatic stress) within an error of about 2% or less as compared to the exact, numerically evaluated, value of cavitation pressure. Two ad-hoc compressibility approximations for cavitation pressure are suggested. These relations, which give very accurate results, appear to provide tight lower and upper bounds on the exact value of cavitation pressure within an error of less than 0.5%.     

Subsonic and transonic flow field in two-dimensional de laval nozzles

This paper presents solutions of subsonic and transonic flow fields in two-dimensional De Laval nozzles with preassinged contraction ration ₁, expansion ration ₂, and throat wall radiusR _*. The effects of the contraction and the expansion angle on nozzle flow, the transformation of flow pattern of a De Laval nozzle in the throat region, and the conditions of occurrence and the governing parameters of the supersonic bubbles are discussed.     

A new high-accuracy scheme for compressible turbulent flows

Shock-capturing and broad-bandwidth scale resolutions are two main challenges of compressible turbulent flow simulation. To meet the rigorous requests, a novel fifth-order hybrid scheme based on a uniform hybrid framework is designed. With the help of a continuous weight operator, the new scheme combines an upwind compact scheme for smooth regions and a compact-reconstruction weighted essentially non-oscillatory scheme for discontinuous regions. Numerical analyses and canonical numerical tests confirm that the new scheme has high accuracy, spectral-like resolution property and shock-capturing capability. Besides, the new scheme shows high computational efficiency compared to the related shock-capturing schemes and hybrid ones.     

An analytical skin friction and heat transfer model for compressible, turbulent, internal flows

An analytical skin friction model for compressible, turbulent, internal, fully developed flow involving adiabatic and non-adiabatic, smooth and rough flows has been developed by extending the incompressible law-of-the-wall relation to compressible cases. The formula recovers Prandtl's incompressible law of friction for pipes (within 2%) for incompressible flow. The model also shows good correlation with available data for compressible, adiabatic flows and flows involving cold wall heat transfer (within 15%). Comparison with hot wall data is only moderate (15–30%). Finally, using Reynold's analogy, the Stanton number and Nusselt numbers may be estimated.     

Shock structure in separated nozzle flows

In the case of high overexpansion, the exhaust jet of the supersonic nozzle of rocket engines separates from nozzle wall because of the large adverse pressure gradient. Correspondingly, to match the pressure of the separated flow region, an oblique shock is generated which evolves through the supersonic jet starting approximately at the separation point. This shock reflects on the nozzle axis with a Mach reflection. Thus, a peculiar Mach reflection takes place whose features depend on the upstream flow conditions, which are usually not uniform. The expected features of Mach reflection may become much difficult to predict, depending on the nozzle shape and the position of the separation point along the divergent section of the nozzle.      

Numerical simulations of compressible flows using multi-fluid models

Numerical simulations of two-fluid flow models based on the full Navier–Stokes equations are presented. The models include six and seven partial differential equations, namely, six- and seven-equation models. The seven-equation model consists of a non-conservative equation for volume fraction evolution of one of the fluids and two sets of balance equations. Each set describes the motion of the corresponding fluid, which has its own pressure, velocity, and temperature. The closure is achieved by two stiffened gas equations of state. Instantaneous relaxation towards equilibrium is achieved by velocity and pressure relaxation terms. The six-equation model is deduced from the seven-equation model by assuming an infinite rate of velocity relaxation. In this model, a single velocity is used for both fluids. The numerical solutions are obtained by applying the Strang splitting technique. The numerical solutions are examined in a set of one, two, and three dimensions for both the six- and seven-equation models. The results indicate very good agreement with the experimental results. There is an insignificant difference between the results of the two models, but the six-equation model is much more economical compared to the seven-equation model.     

Shocklets in compressible flows

The mechanism of shocklets is studied theoretically and numerically for the stationary fluid, uniform compressible flow, and boundary layer flow. The conditions that trigger shock waves for sound wave, weak discontinuity, and Tollmien-Schlichting （T-S） wave in compressible flows are investigated. The relations between the three types of waves and shocklets are further analyzed and discussed. Different stages of the shocklet formation process are simulated. The results show that the three waves in compressible flows will transfer to shocklets only when the initial disturbance amplitudes are greater than the certain threshold values. In compressible boundary layers, the shocklets evolved from T-S wave exist only in a finite region near the surface instead of the whole wavefront.     

Hyperbolicity and one-dimensional waves in compressible two-phase flow models

Hyperbolic models for compressible two-phase flows including a conservative symmetric hyperbolic model are reviewed. The basis for a theory of shock waves is developed within the framework of the latter. The analysis of small amplitude discontinuities allows us to conclude that in general there are two types of shocks corresponding to two sound waves. The problem of transition between a pure phase and a mixture (the phase vacuum problem) is analysed. It is proved that for some models the smooth centred wave solution can not provide such a transition. Within the framework of our conservative model there is the possibility of constructing discontinuous solutions which can resolve the phase vacuum problem.PACS: 47.55Kf, 47.40.-xE. Romenski: On leave from Sobolev Institute of Mathematics, Russian Academy of Sciences, Novosibirsk 630090, Russia     

High-speed visualization and PIV measurements of cavitating flows around a semi-circular leading-edge flat plate and NACA0015 hydrofoil

Cavitating flows around a flat plate with semi-circular leading edge and a NACA0015 hydrofoil at attack angles ranging from 0° to 9° and with varying cavitation number are investigated using high-speed-imaging visualization (HIV) and particle-imaging velocimetry (PIV). Several known types of cavitation common to both foils, but also some different patterns, were observed. At small angles of incidence (less than 3°), cavitation on the plate begins in the form of a streak array (bubble-band) whereas on the hydrofoil as traveling bubbles. For the regimes with developed cavitation on the NACA0015 hydrofoil, the scattered and discontinuous bubble streaks branch and grow but subsequently merge into bubble clouds forming a remarkably regular lattice pattern. Once the incidence angle increased to 9°, the cavitation on the hydrofoil changed to a streaky pattern like that on the plate at small attack angles, whereas the regime on the plate showed no significant changes. The PIV method proved to be usable for measuring the instantaneous velocity also in the gas–vapor phase, albeit with reduced accuracy that was evaluated and accounted for on the basis of the effective (validation-surviving) number of imaging samples. The time-averaged velocity and turbulence moments show that the incipience of cavitation is governed by the development of the carrier-fluid flow around the foil leading edges, but the subsequent flow pattern depends strongly on the cavitation regime displaying markedly different distributions compared to the non-cavitating case. The main cavitation parameters: the maximum cavity length, the cloud cavity streamwise dimensions and the cloud shedding Strouhal number are analyzed and presented in function of the cavitation number and the attack angle in different scaling. The measurements confirm qualitatively the trends reported in the literature, but show also some quantitative differences, notably between the two foils considered.     

Nonisothermal immiscible compressible thermodynamically consistent two-phase flow in porous media

In this paper, we introduce a new model of the nonisothermal immiscible compressible thermodynamically consistent two-phase flow in a porous domain Ω. This model includes the term describing the skeleton and interphase boundary energies. In the framework of the model, we derive the equation for the entropy function in the whole Ω and then obtain the estimate of the maximal entropy of the system.     

Measurement of very small liquid flows

A measuring principle for the flow rate of liquid follows consists in the determination of the volume change of a liquid in a tank in relation to unit of time. The development and the realization of three measurement systems based on this principle are presented in this paper. The devices can measure liquid flows below 0.2 L/h. One flow meter uses an air bubble in the form of a stopper as tracer. A special design keeps the bubble in the measuring tube; it does not have to be replaced. Another flow meter measures the volume changes by determining the pressure at the bottom of a vertical tube. To achieve a continuous operation, a differential design with two corresponding tubes was realized. The third flow meter measures the mass of a drop at the end of a horizontal silicone tube working as a bending beam. The liquid flows through the tube and is forced to form drops at its end. Owing to their design, each of the three devices has distinctive characteristics. A theory of measurement systems that facilitates synthesis and analysis of measuring devices serves as the basis for the development of the flow meters.     

时刻追踪多介质界面运动的动网格方法

在对可压缩多介质流动的数值模拟中,定义介质界面为一种内部边界,由网格的边组成,界面边两侧对应两种不同介质中的网格。通过求解Riemann问题追踪介质界面上网格节点的运动,同时采用局部重构的动网格技术处理介质界面的大变形问题,并将介质界面定义为网格变形边界,以防止该边界上网格体积为负。运用HLLC格式求解ALE方程组得到整个多介质流场的数值解。最后从几个多介质流模型的计算结果可以看出,本文的动网格方法是可行的,而且可以时刻追踪介质界面的运动状态。     

On formulas for the Rayleigh wave velocity in pre-stressed compressible solids

In this paper, formulas for the velocity of Rayleigh waves in compressible isotropic solids subject to uniform initial deformations are derived using the theory of cubic equation. They are explicit, have simple algebraic forms, and hold for a general strain energy function. Unlike the previous investigations where the derived formulas for Rayleigh wave velocity are approximate and valid for only small enough values of pre-strains, this paper establishes exact formulas for Rayleigh wave velocity being valid for any range of pre-strains. When the prestresses are absent, the obtained formulas recover the Rayleigh wave velocity formula for compressible elastic solids. Since obtained formulas are explicit, exact and hold for any range of pre-strains, they are good tools for evaluating nondestructively prestresses of structures.     

Supersonic flow separation in planar nozzles

We present experimental results on separation of supersonic flow inside a convergent–divergent (CD) nozzle. The study is motivated by the occurrence of mixing enhancement outside CD nozzles operated at low pressure ratio. A novel apparatus allows investigation of many nozzle geometries with large optical access and measurement of wall and centerline pressures. The nozzle area ratio ranged from 1.0 to 1.6 and the pressure ratio ranged from 1.2 to 1.8. At the low end of these ranges, the shock is nearly straight. As the area ratio and pressure ratio increase, the shock acquires two lambda feet. Towards the high end of the ranges, one lambda foot is consistently larger than the other and flow separation occurs asymmetrically. Downstream of the shock, flow accelerates to supersonic speed and then recompresses. The shock is unsteady, however, there is no evidence of resonant tones. The separation shear layer on the side of the large lambda foot exhibits intense instability that grows into large eddies near the nozzle exit. Time-resolved wall pressure measurements indicate that the shock oscillates in a piston-like manner and most of the energy of the oscillations is at low frequency.