Fluid Of Finite Thickness Research Articles

Krauklis wave propagation within a complex fracture system: Modeling via a two-dimensional time-harmonic boundary element method.

Fluid-filled fractures involving kinks and branches result in complex interactions between Krauklis waves-highly dispersive and attenuating pressure waves within the fracture-and the body waves in the surrounding medium. For studying these interactions, we introduce an efficient 2D time-harmonic elastodynamic boundary element method. Instead of modeling the domain within a fracture as a finite-thickness fluid layer, this method employs zero-thickness, poroelastic Linear-Slip Interfaces to model the low-frequency, local fluid-solid interaction. Using this method, the scattering of Krauklis waves by a single kink along a straight fracture and the radiation of body waves generated by Krauklis waves within complex fracture systems are examined.

Magnetohydrodynamic Kelvin–Helmholtz instability for finite-thickness fluid layers

We have derived the analytical formulas for the Kelvin–Helmholtz instability (KHI) of two superposed finite-thickness fluid layers with the magnetic field effect into consideration. The linear growth rate of KHI will be reduced when the thickness of the fluid with large density is decreased or the thickness of fluid with small density is increased. When the thickness and the magnetic field act together on the KHI, the effect of thickness is more obvious when the magnetic field intensity is weak. The magnetic field transition layer destabilizes (enforces) the KHI, especially in the case of small thickness of the magnetic field transition layer. When considering the effect of magnetic field, the linear growth rate of KHI always decreases after reaching the maximum with the increase of total thickness. The stronger the magnetic field intensity is, the more obvious the growth rate decreases with the total thickness. Thus, it should be included in applications where the effect of fluid thickness on the KHI cannot be ignored, such as in double-cone ignition scheme for inertial confinement fusion.

The phase effect on the Richtmyer–Meshkov instability of a fluid layer

Shock-induced finite-thickness fluid layer evolution is investigated numerically and theoretically. Specifically, two-dimensional helium layers consisting of two interfaces owning diverse perturbation phases are considered to explore the interface-coupling on the Richtmyer–Meshkov instability (RMI). A general linear model is first established to quantify the phase effect on the RMI of the two interfaces of an arbitrary fluid layer. The linear model is validated with the present numerical results. As the phase difference between the two interfaces' perturbations increases, the linear amplitude growth rates of the two interfaces are larger. The influences of diverse parameters on the interface-coupling are concerned. Moreover, the nonlinearity of the RMI of the two interfaces is dependent on the phase difference. Finally, spectrum analysis is performed to investigate the phase effect on perturbation growths of the first three-order harmonics of the two interfaces.

Effect of initial phase on the Rayleigh–Taylor instability of a finite-thickness fluid shell

Rayleigh–Taylor instability (RTI) of finite-thickness shell plays an important role in deep understanding the characteristics of shell deformation and material mixing. The RTI of a finite-thickness fluid layer is studied analytically considering an arbitrary perturbation phase difference on the two interfaces of the shell. The third-order weakly nonlinear (WN) solutions for RTI are derived. It is found the main feature (bubble-spike structure) of the interface is not affected by phase difference. However, the positions of bubble and spike are sensitive to the initial phase difference, especially for a thin shell (kd < 1), which will be detrimental to the integrity of the shell. Furthermore, the larger phase difference results in much more serious RTI growth, significant shell deformation can be obtained in the WN stage for perturbations with large phase difference. Therefore, it should be considered in applications where the interface coupling and perturbation phase effects are important, such as inertial confinement fusion.

Kelvin–Helmholtz instability of two finite-thickness fluid layers with continuous density and velocity profiles

The effect of density and velocity gradients on the Kelvin–Helmholtz instability (KHI) of two superimposed finite-thickness fluid layers are analytically investigated. The linear normalized frequency and normalized growth rate are presented. Then, their behavior as a function of the density ratio of the light fluid to the heavy one (r) was analyzed and compared to the case of two semi-infinite fluid layers. The results showed that the values of normalized frequency of KHI for two finite-thickness fluid layers are less than their counterparts for two semi-infinite fluid layers. The behavior of normalized growth rate as a function of the velocity and density gradients capitulates to the effect of velocity gradient at the large values of (r).

Thin layer model for nonlinear evolution of the Rayleigh-Taylor instability

On the basis of the thin layer approximation [Ott, Phys. Rev. Lett. 29, 1429 (1972)], a revised thin layer model for incompressible Rayleigh-Taylor instability has been developed to describe the deformation and nonlinear evolution of the perturbed interface. The differential equations for motion are obtained by analyzing the forces (the gravity and pressure difference) of fluid elements (i.e., Newton's second law). The positions of the perturbed interface are obtained from the numerical solution of the motion equations. For the case of vacuum on both sides of the layer, the positions of the upper and lower interfaces obtained from the revised thin layer approximation agree with that from the weakly nonlinear (WN) model of a finite-thickness fluid layer [Wang et al., Phys. Plasmas 21, 122710 (2014)]. For the case considering the fluids on both sides of the layer, the bubble-spike amplitude from the revised thin layer model agrees with that from the WN model [Wang et al., Phys. Plasmas 17, 052305 (2010)] and the expanded Layzer's theory [Goncharov, Phys. Rev. Lett. 88, 134502 (2002)] in the early nonlinear growth regime. Note that the revised thin layer model can be applied to investigate the perturbation growth at arbitrary Atwood numbers. In addition, the large deformation (the large perturbed amplitude and the arbitrary perturbed distributions) in the initial stage can also be described by the present model.

Propagation characteristics of pseudo-Scholte waves at the interface between finite-thickness fluid layer and quasi-saturated porous half-space

The propagation of interface waves at the interface between a fluid-saturated porous medium and a fluid has been extensively investigated in the last three decades due to its various and wide applications in several fields including earthquake engineering and materials testing. Although the sea floor is usually covered with porous marine sediment, the previous interface wave theories are rarely used for submarine acoustic problems for the following reasons. 1) In addition to hard porous media, unconsolidated soft porous media exist widely in the seabed, which are seldom considered in previous studies. 2) The depth of seawater is limited, and in many cases it cannot be regarded as a half-space. 3) The fluid-saturated porous medium model cannot describe the effect of a small number of bubbles caused by decomposition of organic matter in the sediment. Hence, the present paper focuses on the low-frequency pseudo-Scholte waves at the interface between an overlying fluid layer of finite thickness and a quasi-saturated porous half-space. The overlying fluid is assumed to be ideal compressible water and the quasi-saturated porous media are assumed to be sandstone and unconsolidated sediment and modeled by Biot theory. A fluid equivalent model is used to analyze the effects of the bubbles in the pores. Based on the boundary conditions, the closed-form dispersion equations of far-field interface waves are derived by using classical potential function method. The velocity and attenuation of pseudo-Scholte wave are determined by Newton iteration in a reasonable rooting interval. The analytical expressions of the displacement field and fluid pressure distribution caused by pseudo-Scholte waves are also derived. Then, based on the derived theoretical formulation, the numerical examples of calculations are presented. Our calculation results show that the stiffness of porous medium significantly affects the mode, phase velocity, displacement and fluid pressure distribution of interface waves; the phase velocity of the pseudo-Scholte wave in the finite-thickness fluid/fluid-saturated porous half-space is related to the ratio of the wavelength to the thickness of the fluid layer; the phase velocity of the shear wave is insensitive to a small number of bubbles dissolved in the pores, but the existence of bubbles has a significant influence on the phase velocity of the compressional wave and the pseudo-Scholte wave. Furthermore, the existence of bubbles can significantly affect the distribution of the pore pressure.

Nonlinear saturation of Rayleigh-Taylor instability in a finite-thickness fluid layer

Nonlinear amplitude saturation (NAS) of the fundamental mode of Rayleigh–Taylor instability (RTI) in a finite-thickness incompressible fluid layer is investigated analytically by considering high-order corrections (HOCs) up to the ninth order. The results of classical RTI [Liu et al., Phys. Plasmas 19, 042705 (2012)] can be recovered for the normalized fluid thickness kd→∞. It is found that the NAS of the fundamental mode on the lower and upper interfaces is clearly larger than its third-order counterpart [Wang et al., Phys. Plasmas 21, 122710 (2014)] when the HOCs are considered, especially for the lower (linearly unstable) interface. Furthermore, the NAS on both interfaces exhibits the trend of convergence with increasing order of corrections.

Linear Growth of Rayleigh–Taylor Instability of Two Finite-Thickness Fluid Layers**Supported by the National Natural Science Foundation of China under Grant Nos 11275031, 11475034, 11575033 and 11274026, and the National Basic Research Program of China under Grant No 2013CB834100.

The linear growth of Rayleigh–Taylor instability (RTI) of two superimposed finite-thickness fluids in a gravitational field is investigated analytically. Coupling evolution equations for perturbation on the upper, middle and lower interfaces of the two stratified fluids are derived. The growth rate of the RTI and the evolution of the amplitudes of perturbation on the three interfaces are obtained by solving the coupling equations. It is found that the finite-thickness fluids reduce the growth rate of perturbation on the middle interface. However, the finite-thickness effect plays an important role in perturbation growth even for the thin layers which will cause more severe RTI growth. Finally, the dependence of the interface position under different initial conditions are discussed in some detail.

Evolution of perturbations of a charged interface between immiscible inviscid fluids in the interelectrode gap

Perturbations of the interface between two immiscible ideal fluids of finite thickness (the lower and upper fluids are the conductor and the dielectric, respectively) located in the gap between two electrodes are considered. In the cases of the “shallow” and “deep” upper fluid the dispersion relations of linear waves and their longwave expansions are found. The methods of determining the space-time evolution of an initial surface perturbation are developed on the basis of the linear approximation. In the cases of the “shallow” and “deep” upper fluid examples of the development of an initial perturbation of the “step” type are given. The development of an initial perturbation of the “step” type are also considered in the near-critical electric fields and in the case of degeneration of cubic dispersion.

Surface instability in a finite thickness fluid saturated porous layer

The Rayleigh-Taylor (RT) instability at the interface between fluid and fluid saturated sparsely packed porous medium has been investigated making use of boundary layer approximation and Saffmann [8] boundary condition. An analytical solution for dispersion relation is obtained and is numerically evaluated for different values of the parameters. It is shown that RT instability can be controlled by a suitable choice of the thickness of porous layer, ratio of viscosities and the slip parameter.

Numerical simulations of Richtmyer–Meshkov instabilities in finite-thickness fluid layers

Direct numerical simulations of Richtmyer–Meshkov instabilities in shocked fluid layers are reported and compared with analytic theory. To investigate new phenomena such as freeze-out, interface coupling, and feedthrough, several new configurations are simulated on a two-dimensional hydrocode. The basic system is an A/B/A combination, where A is air and B is a finite-thickness layer of freon, SF6, or helium. The middle layer B has perturbations either on its upstream or downstream side, or on both sides, in which case the perturbations may be in phase (sinuous) or out of phase (varicose). The evolution of such perturbations under a Mach 1.5 shock is calculated, including the effect of a reshock. Recently reported gas curtain experiments [J. M. Budzinski et al., Phys. Fluids 6, 3510 (1994)] are also simulated and the code results are found to agree very well with the experiments. A new gas curtain configuration is also considered, involving an initially sinuous SF6 or helium layer and a new pattern, opposite mushrooms, is predicted to emerge. Upon reshock a relatively simple sinuous gas curtain is found to evolve into a highly complex pattern of nested mushrooms.

Rayleigh–Taylor and Richtmyer–Meshkov instabilities in finite-thickness fluid layers

Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities are considered in a fluid layer of thickness t having perturbations of arbitrary wave number k at either one or both interfaces. The evolution of the perturbation amplitudes η1,2(τ), τ =time, is given analytically in terms of a coupling angle θ which measures the strength of the coupling between interfaces 1 and 2. A new type of freeze-out in shocked layers is reported according to which, the proximity of the two interfaces can, under proper conditions, lead to the complete freeze-out of one, but not both, perturbations. For example, to freeze the first interface one needs η1(0)/η2(0)= sin θ. Freeze-out cannot be achieved in the RT case; instead, one can kill one of the modes. For example, setting η1(0)/η2(0)= tan(θ/2) will kill the exponentially growing mode, leaving only the oscillatory mode at both interfaces.

Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble fronts

A potential flow model of Rayleigh–Taylor and Richtmyer–Meshkov bubbles on an interface between an incompressible fluid and a constant supporting pressure (Atwood number A=1) is presented. In the model, which extends the work of Layzer [Astrophys. J. 122, 1 (1955)], ordinary differential equations for the bubble heights and curvatures are obtained by considering the potential flow equations near the bubble tips. The model is applied to two-dimensional single-mode evolution as well as two-bubble competition, for both the Rayleigh–Taylor (RT) and the Richtmyer–Meshkov (RM) instabilities, the latter treated in an impulse approximation. The model predicts that the asymptotic velocity of a single-mode RM bubble of wavelength λ decays as λt−1, in contrast with the constant asymptotic velocity attained in the RT case. Bubble competition, which is believed to determine the multimode front evolution, is demonstrated for both the RT and RM instabilities. The capability of the model to predict bubble growth in a finite-thickness fluid layer is shown. Finally, the model is applied to the evolution of three-dimensional modes with an initial rectangular geometry. The model yields the aspect ratio dependence of the early nonlinear stages, in agreement with third-order perturbation theory. However, in the late nonlinear stage, the model predicts that the bubbles forget the initial geometry and attain the fastest growing shape, with a circular tip. The model results are in good agreement with full hydrodynamic simulations and analytic solutions, where available.

Nonlinear Kelvin—Helmholtz instability in magnetic fluids of finite thickness: effects of a tangential magnetic field

The nonlinear stability of a horizontal interface separating two streaming magnetic fluids of finite thickenss is investigated in two dimensions. The fluids are considered to be inviscid and incompressible. The magnetic field is applied along the direction of streaming. The method of multiple scales, in both space and time, is used to examine the stability properties of the system arising from second-harmonic resonance. A pair of partial differential equations for the amplitude of the wave and its second harmonic are derived. These describe the evolution of the wave train up to cubic order, and may be regarded as the counterparts of the single nonlinear Schrödinger equation that occurs in the non-resonant case. The stability condition of this equation is discussed both analytically and numerically, and stability diagrams are obtained. Regions of stability and instability are identified. The nonlinear cut-off wavenumber separating the regions of stability from those of instability is obtained. The equation governing the evolution of the amplitude at the critical point is also obtained, which leads to a nonlinear Klein—Gordon equation.

Nonlinear Rayleigh-Taylor instability in magnetic fluids between two parallel plates

A nonlinear stage of the two-dimensional Rayleigh-Taylor instability for two magnetic fluids of finite thickness is studied by including the effect of surface tension between the two fluids. The system is subjected to a tangential magnetic field. The method of multiple scale perturbations is used in order to obtain uniformly valid expansions near the cutoff wavenumber separating stable and unstable deformations. Two nonlinear Schrodinger equations are obtained, one of which leads to the determination of the cutoff wavenumber. The other Schrodinger equation is used to analyze the stability of the system. It is found that if a finite-amplitude disturbance is stable, then a small modulation to the wave is also stable. It is also found that the tangential magnetic field plays a dual role in the stability criterion. Finally, the magnetic permeability constants of the fluid affect the stability conditions.

Instability of a system of horizontal layers of immiscible fluids heated from above

The stability of the equilibrium is studied in a system heated from above and consisting of two layers of different fluids of finite thickness. The physical mechanism of the instability is discussed and shown to be very different from the Rayleigh mechanism. The ranges of variation of the parameters of the system in which the effect is possible are found, and the boundaries of stability are determined quantitatively.

Theory of the drift of asymmetrically-heated lithospheric plates

An approximate solution to the problem of the drift of a raft formed of two parallel wires, differentially heated, rigidly coupled and floating in a fluid of finite thickness and with linear viscosity, has been obtained and is shown to agree well with experiment up to a multiplicative constant. In addition, the solution to the problem of the drift of a solid raft with a suspended single-wire heater also shows good agreement with experiment up to a multiplicative constant. In both cases, the rafts drift with constant velocity. For small amounts of heat, the drift velocity is proportional to the first power of the heat input; for large amounts of heat, the drift velocity is proportional to the square root of the heat input. Within imponderable factors of an order of magnitude, the drift velocities are appropriate for drift of lithospheric plates containing both an oceanic and a continental part.