Incompressible Theory Research Articles

The unsteady Kutta condition on an airfoil in a surging flow

This work presents an experimental validation study of Isaacs’ incompressible unsteady-airfoil theory at Reynolds numbers above $10^{6}$ , and explores the validity of the classical Kutta condition applied to surging flows. Harmonic variation of the free-stream velocity was produced by rotating choke vanes in an unsteady transonic wind tunnel, with time-resolved lift coefficients determined from surface pressure measurements on a NACA 0018 airfoil. Unsteady lift results demonstrate the same trends with reduced frequency and velocity amplitude ratio that are predicted by Isaacs’ theory. However, significant deviations of the lift magnitude and phase angle are observed. In order to understand the cause of these deviations, the background-oriented schlieren technique was used to visualize density gradients in the immediate vicinity of the airfoil trailing edge. The time-resolved background-oriented schlieren displacement field indicates oscillatory behaviour of the trailing-edge stagnation streakline, which violates the classical Kutta condition for this unsteady surging flow.

Compressible Fanno flows in micro-channels: An enhanced quasi-2D numerical model for turbulent flows

Fanno theory provides an analytical model for one-dimensional confined viscous compressible flows. The model holds under the assumptions of adiabatic flow and constant cross-section channel. From theory, the differential of every flow-related quantity is expressed as a function of Mach number and friction factor. One-dimensional flow numerical models can be derived by discretizing Fanno equations. However, theory does not assess how to evaluate friction, while the model works properly only if friction is estimated correctly. Compressibility and turbulence act by deforming the velocity profile making it flatter. Assuming the friction factor function of the Reynolds number alone, in line with incompressible flow theory, is thus not correct. Better correlations should include the Mach number to address compressibility effects. Here, the impact of turbulence and compressibility on the velocity profiles in a micro-channel is analysed by means of CFD simulations. Friction factor correlations are deduced for turbulent micro-flows. The impact of the velocity profile on other quantities, such as dynamic pressure and bulk temperature, needed for the numerical model operation, is also evaluated. Additional correlations for these quantities overcome the instrinsic limits of the one-dimensional model, necessarily unaware of local velocity profiles, in a quasi-2D fashion significantly improving its predicting capabilities.

Penetration Behavior of High-Density Reactive Material Liner Shaped Charge.

The traditional polytetrafluoroethylene (PTFE)/Al reactive material liner shaped charge generally produces insufficient penetration depth, although it enlarges the penetration hole diameter by chemical energy release inside the penetration crater. As such, a novel high-density reactive material liner based on the PTFE matrix was fabricated, and the corresponding penetration performance was investigated. Firstly, the PTFE/W/Cu/Pb high-density reactive material liner was fabricated via a cold pressing/sintering process. Then, jet formation and penetration behaviors at different standoffs were studied by pulse X-ray and static experiments, respectively. The X-ray results showed that the PTFE/W/Cu/Pb high-density reactive material liner forms an excellent reactive jet penetrator, and the static experimental results demonstrated that the penetration depth of this high-density reactive jet increased firstly and then decreased by increasing the standoff. When the standoff was 1.5 CD (charge diameter), the penetration depth of this reactive jet reached 2.82 CD, which was significantly higher than that of the traditional PTFE/Al reactive jet. Moreover, compared with the conventional metal copper jet penetrating steel plates, the entrance hole diameter caused by this high-density reactive jet improved 29.2% at the same standoff. Lastly, the chemical reaction characteristics of PTFE/W/Cu/Pb reactive materials were analyzed, and a semi-empirical penetration model of the high-density reactive jet was established based on the quasi-steady ideal incompressible fluid dynamics theory.

Free fall water entry of a wedge tank into calm water in three degrees of freedom

The hydrodynamic problem of a two dimensional wedge tank filled with liquid entering a calm water surface is analysed based on the incompressible velocity potential theory. The motion effect of inner liquid on the entry process is investigated through comparison with the result containing equivalent solid mass or the liquid being frozen. The problem is solved through the boundary element method in the time domain. Two separated computational regions are constructed. One is the inner domain for the internal liquid, and the other is the outer open domain for the open water. The former is solved in the physical coordinate system, and the latter is solved in a stretched coordinate system. The solutions of two separated domains are connected through the motion of the body. The auxiliary function method is extended to decouple the nonlinear mutual dependence between fluid loads from two separated domains and the body motion. Detailed results for wedge motion, external impact pressure and free surface, and for internal pressure, free surface deformation and liquid motion are provided. Through comparison with the results of a wedge tank with frozen ice, in-depth discussion on the effect of the inner liquid is provided.

Nonlinear behaviour of convergent Richtmyer–Meshkov instability

A novel shock tube is designed to investigate the nonlinear feature of convergent Richtmyer–Meshkov instability on a single-mode interface formed by a soap film technique. The shock tube employs a concave–oblique–convex wall profile which first transforms a planar shock into a cylindrical arc, then gradually strengthens the cylindrical shock along the oblique wall, and finally converts it back into a planar one. Therefore, the new facility can realize analysis on compressibility and nonlinearity of convergent Richtmyer–Meshkov instability by eliminating the interface deceleration and reshock. Five sinusoidal$\text{air}{-}\text{SF}_{6}$interfaces with different amplitudes and wavelengths are considered. For all cases, the perturbation amplitude experiences a linear growth much longer than that in the planar geometry. A compressible linear model is derived by considering a constant uniform fluid compression, which shows a slight difference to the incompressible theory. However, both the linear models overestimate the perturbation growth from a very early stage due to the presence of strong nonlinearity. The nonlinear model of Wanget al.(Phys. Plasmas, vol. 22, 2015, 082702) is demonstrated to predict well the amplitude growth up to a normalized time of 1.0. The prolongation of the linear increment is mainly ascribed to the counteraction between the promotion by geometric convergence and the suppression by nonlinearity. Growths of the first three harmonics, obtained by a Fourier analysis of the interface contour, provide a first thorough validation of the nonlinear theory.

Finite element analysis of aero-hydroelastic stability of arbitrary shape panels

A fluid-structure model based on Mindlin-Reissner plate theory and linearized incompressible potential flow theory is used to study the aero-hydroelastic stability of panels of arbitrary planforms. A finite-element procedure is developed to reduce the continuous system to a finite flow-structure system, which is solved in the frequency domain. The finite element method is crucial for the study of panels of arbitrary geometry and circumvents difficulties with the assumed mode shape approach. This paper is the first finite-element formulation of a fully coupled fluid-structure model of the aero-hydroelastic panel stability problem for incompressible flow. Circular, square, and triangular panels surrounded by a rigid baffle with different boundary conditions are studied. Eigenvalues of the system are surveyed as a function of the hydrodynamic loading at high and low density ratios. Critical flow velocity at divergence-onset and modal coalescence divergence or flutter are determined and compared for the three planforms. For simply supported or clamped panels of equal area and flexural rigidity, as the fluid velocity increases from zero the circular panel diverges first then the square and equilateral triangle, the later is the most stable. In order to compare the relative stability of the three geometries considered, a velocity ratio that consolidates divergence and flutter conditions for the three panels is introduced. Such a parameter offers a simple formula for the divergence velocity in terms of the panel geometry and flow and material properties, and is very useful to the designer in a wide range of engineering applications.

Water Entry of A Wedge Into Waves in Three Degrees Offreedom

Abstract The hydrodynamic problem of a two-dimensional wedge entering into a nonlinear wave in three degrees of freedom is investigated based on the incompressible velocity potential theory. The problem is solved through the boundary element method in the time domain. To avoid numerical difficulties due to an extremely small contact area at the initial stage, a stretched coordinate system is used based on the ratio of the Cartesian system in the physical space to the distance travelled by the wedge in the vertical direction. The mutual dependence of body motion and wave loading is decoupled by using the auxiliary function method. Detailed results about body accelerations, velocities and displacements at different Froude numbers or different waves are provided, and the mutual effect between body motion and wave loading is analysed in depth.

Free fall motion of a floating body: Bubble formation and its effect

Abstract The hydrodynamic problem of a floating body moving downwards into water with prescribed velocity or in free fall motion is solved based on the incompressible velocity potential theory through the boundary element method in the time domain. In particular, the focus is on the whole process of flow detachment, the formation of an open cavity, closure of the cavity or formation of entrapped air bubble, collision of the inner jet with the body surface and jet impact with free surface. The whole problem is divided into several stages and methodology is introduced to resolve the numerical challenge at each stage. In particular the dual coordinate system method is adopted to resolve the local sharp spatial and temporal variation in the impact zone, and the domain decomposition method with Riemann second sheet is adopted to resolve domain overlapping. An auxiliary function method is used to decouple the nonlinear mutual dependence of fluid loading, body motion and bubble deformation. Simulations are undertaken for a floating body moving into water at constant velocity or constant acceleration, and in free motion, respectively. Detailed results for pressure, free surface profile, bubble deformation and body motion are provided, and their physical implications are discussed.

Water entry of an expanding body with and without splash

A self-similar flow generated by water entry of an expanding two-dimensional smooth and curved body is studied based on the incompressible velocity potential theory, with gravity and surface tension effects being ignored. At each expansion speed, mathematical solutions for detached flow with a splash jet and attached flow with jet leaning on the body surface are obtained, both of which have been found possible in experiment, corresponding to hydrophobic and hydrophilic bodies, respectively. The problem is solved using the integral hodograph method, which converts the governing Laplace equations into two integral equations along the half real and imaginary axes of a parameter plane. For the detached flow, the conditions for continuity and finite spatial derivative of the velocity at the point of flow departing form the body surface are imposed. It is found that the Brillouin–Villat criterion for flow detachment of steady flow is also met in this self-similar flow, which requires the curvatures of the free surface and the body surface to be the same at the detachment point. Solutions for the detached flow have been obtained in the whole range of the expansion speeds, from zero to infinity, relative to the water entry speed. For the attached flow there is a minimal expansion speed below which no solution cannot be obtained. Detailed results in terms of pressure distribution, free surface shape and streamlines and tip angle of the jet are presented. It is revealed that when solutions for both detached and attached flows exist, the pressure distributions on the cylinder surface are almost the same up to the point near the jet root. Beyond that point, the pressure relative to the ambient one drops to zero at the detachment point in the former, while it drops below zero in the jet attached on the body and then returns to zero at the contact point in the latter.

Compressible Fanno flows in micro-channels: An enhanced quasi-2D numerical model for laminar flows

Micro-scale fluid systems are becoming common in many applications ranging from electronic cooling to refrigeration systems and more. One-dimensional numerical models represent a simple and fast tool for the design of such devices, yet they struggle to accurately predict the flow characteristics in compressible micro-flows. Under the adiabatic assumption, the elegant theory developed by Fanno allows models for the viscous compressible flow in constant cross-section channels to be easily built. Although reasonably accurate, these models suffer from drawbacks inherent to their being one-dimensional, as such they cannot take into account the local profiles of quantities like the velocity and the temperature. In cascade, this results into incorrect evaluations of other dependent quantities, such as the dynamic pressure and the fluid thermophysical properties. The mismatch turns large when the fluid compressibility becomes important. As the Mach number grows, the velocity profile changes, and so the friction factor, even though a reliable model for predicting this change is still missing. In fact, a constant friction factor throughout the channel is generally assumed, following the incompressible flow theory. Here, a set of correlations is proposed improving the 1D theory accuracy by taking into account the effects of the non-uniform velocity and temperature profiles in a quasi-2D fashion. A detailed analysis of the velocity profiles at different Mach numbers coming from a large set of CFD simulations results in a model for assessing the impact of compressibility on friction and other quantities. The numerical model proposed, being able to properly account for the compressibility effects, offers an improved tool for the design of micro-scale fluid systems. Extending the analysis to include heat transfer is not difficult as the effect of heat flux will be analogous to the effect of pressure drop due to friction.

Moderation of flame acceleration in obstructed cylindrical pipes due to gas compression

The role of gas compression in the process of extremely fast flame acceleration in obstructed cylindrical tubes is studied analytically and validated by computational simulations. This acceleration is associated with a powerful jet-flow produced by delayed burning in spaces between the obstacles. The acceleration mechanism is Reynolds-independent and conceptually laminar, with turbulence playing only a supplementary role. In this work, the incompressible formulation is extended to account for small but finite initial flame Mach number up to the first-order terms. As a result, it is shown that flame-generated compression mitigates the acceleration process noticeably. Being qualitatively the same as in two-dimensional channels, quantitatively, the effect is much stronger in cylindrical pipes, which reduces the validity domain of the incompressible cylindrical-axisymmetric formulation as compared to the incompressible two-dimensional theory. The latter is also tested when the theory is validated by the computational simulations.

Breaking wave impact on a floating body with air bubble effect

The hydrodynamic problem of a breaking wave impacting on a floating body with air bubble effect is modelled based on the incompressible velocity potential theory, which is solved using the boundary element method in the time domain. To avoid the numerical inaccuracies due to the sharp temporal and spatial variations of velocity and pressure at the initial stage of impact, a dual system is adopted. The simulation close to the impact zone is conducted in a stretched coordinate system, while away from the impact zone the deformation and propagation of incoming overturning wave is simulated in the physical coordinate system. The continuities of both pressure and velocity are enforced at the interface of two zones. When the impact zone is no longer small, the dual systems will be merged and the simulation will be undertaken in one single domain. The air bubble trapped between the breaking wave and the solid surface is taken into account based on the assumption that the trapped air undergoes an adiabatic process. An auxiliary function method is used to decouple the nonlinear mutual dependence of fluid loading, body motion and bubble deformation. Simulations are undertaken for cases related to breaking wave impact in various engineering problems, including a solid coastal wall, a freely floating ship cross section, a floating breakwater and a tension leg platform. Detailed results for pressure, free surface profile, bubble deformation and body motion are provided, and their physical implications are discussed.

Unit Reynolds number, Mach number and pressure gradient effects on laminar\u2013turbulent transition in two-dimensional boundary layers

The influence of unit Reynolds number (Re_1=17.5times 10^{6}–80times 10^{6},text {m}^{-1}), Mach number (M= 0.35–0.77) and incompressible shape factor (H_{12} = 2.50–2.66) on laminar–turbulent boundary layer transition was systematically investigated in the Cryogenic Ludwieg-Tube Göttingen (DNW-KRG). For this investigation the existing two-dimensional wind tunnel model, PaLASTra, which offers a quasi-uniform streamwise pressure gradient, was modified to reduce the size of the flow separation region at its trailing edge. The streamwise temperature distribution and the location of laminar–turbulent transition were measured by means of temperature-sensitive paint (TSP) with a higher accuracy than attained in earlier measurements. It was found that for the modified PaLASTra model the transition Reynolds number (Re_{text {tr}}) exhibits a linear dependence on the pressure gradient, characterized by H_{12}. Due to this linear relation it was possible to quantify the so-called ‘unit Reynolds number effect’, which is an increase of Re_{text {tr}} with Re_1. By a systematic variation of M, Re_1 and H_{12} in combination with a spectral analysis of freestream disturbances, a stabilizing effect of compressibility on boundary layer transition, as predicted by linear stability theory, was detected (‘Mach number effect’). Furthermore, two expressions were derived which can be used to calculate the transition Reynolds number as a function of the amplitude of total pressure fluctuations, Re_1 and H_{12}. To determine critical N-factors, the measured transition locations were correlated with amplification rates, calculated by incompressible and compressible linear stability theory. By taking into account the spectral level of total pressure fluctuations at the frequency of the most amplified Tollmien–Schlichting wave at transition location, the scatter in the determined critical N-factors was reduced. Furthermore, the receptivity coefficients dependence on incidence angle of acoustic waves was used to correct the determined critical N-factors. Thereby, a found dependency of the determined critical N-factors on H_{12} decreased, leading to an average critical N-factor of about 9.5 with a standard deviation of sigma approx 0.8.

The finite cell method for nearly incompressible finite strain plasticity problems with complex geometries

In this paper, the performance of the Finite Cell Method is studied for nearly incompressible finite strain plasticity problems. The Finite Cell Method is a combination of the fictitious domain approach with the high-order Finite Element Method. It provides easy mesh generation capabilities for highly complex geometries; moreover, this method offers high convergence rates, the possibility to overcome locking and robustness against high mesh distortions. The performance of this method is numerically investigated based on computations of benchmark and applied problems. The results are also verified with the h- and p-version Finite Element Method. It is demonstrated that the Finite Cell Method is an appropriate simulation tool for large plastic deformations of structures with complex geometries and microstructured materials, such as porous and cellular metals that are made up of ductile materials obeying nearly incompressible J2 theory of plasticity.

Numerical Study of Transitional SD7003 Airfoil Interacting with Canonical Upstream Flow Disturbances

The current work reports on the results of high-accuracy two-dimensional and high-fidelity three-dimensional Navier–Stokes simulations of an SD7003 airfoil interacting with canonical upstream flow disturbances at low and moderate Reynolds numbers corresponding to laminar and transitional flow regimes. Three deterministic forms of upstream flow disturbances are considered, including sharp-edge gust, time-harmonic gust, and Taylor vortex models. The results obtained for the airfoil unsteady aerodynamic responses to gust perturbations with variable amplitude and duration are discussed in comparison with results from linearized inviscid incompressible unsteady aerodynamic theory to elucidate the importance of viscous effects in the considered flow regimes. Overall, the investigation reveals that the high-amplitude gust responses correlate well with inviscid theories, which is in contrast to the low-amplitude gust responses that appear to be strongly affected by viscous effects.

Cosmic Ray Diffusion Tensor throughout the Heliosphere Derived from a Nearly Incompressible Magnetohydrodynamic Turbulence Model

Abstract The radial and rigidity dependence of the cosmic ray (CR) diffusion tensor is investigated on the basis of a recently developed 2D and slab turbulence transport model using nearly incompressible theory. We study CR diffusion coefficients in two regions: 0.29 to 1 au, and 1 to 75 au. In the former case, we use 2D and radial slab turbulence transport models, and in the latter case, 2D and perpendicular slab turbulence transport models. We employ quasi-linear theory and nonlinear guiding center theory, respectively, to determine the parallel and perpendicular elements of the CR diffusion tensor. We also present the effect of both weak and moderately strong turbulence on the drift element of the CR diffusion tensor. We find that in the solar wind ecliptic plane (1) the radial mean free path (mfp) is dominated by diffusion parallel to the mean magnetic field and is nearly constant from 0.29 to 1 au; (2) from 1 to 75 au, the role of the perpendicular mfp becomes increasingly important despite the parallel mfp being about three orders of magnitude larger than the perpendicular mfp. The radial mfp initially decays slowly and then faster after ∼6 au; (3) beyond ∼10 au, pickup-ion-driven turbulence is the most important factor in determining the various mean free paths since stream interactions weaken with increasing heliocentric distance; (4) the rigidity (P) dependence of the parallel mfp is proportional to from 10 to 103 MV, but in the distant heliosphere, its dependence increases for higher rigidities and is proportional to . In contrast, the perpendicular mfp is weakly influenced by CR rigidity; (5) the drift length scale is comparable to the perpendicular mfp beyond ∼10 au; and (6) strong turbulence may introduce a new “drift” component, the direction of which is normal to the mean magnetic field.

The dynamic behaviors of a bubble in a confined domain

The oscillation of a bubble near the free surface confined by a cylindrical tank is simulated by the axisymmetric boundary integral method (BIM) code. Firstly, the axis-symmetrical numerical modeling is established based on the incompressible potential flow theory, which is followed by the validation with mechanism experiments carried out using the spark-generated bubble equipment and its auxiliary system, e.g., the high-speed camera. Good agreements are achieved between the numerical results and the experimental ones. Afterwards, the dynamics of a bubble near the free surface in a cylindrical tank and its induced pressure on the tank wall are investigated. Due to the existence of tank solid wall, the water skirt is advanced to form at the bubble expansion stage rather than at the bubble toroidal stage for the case of no tank. In addition, the effects of tank radius on the bubble motion are discussed and a critical radius for negligible effects is obtained. Finally, the bursting at the free surface and the exposure at the bottom induced by the bubble are also studied to provide a reference for underwater explosions of the shallow or the tank experiments.

Nonlinear internal wave at the interface of two-layer liquid due to a moving hydrofoil

This paper is concerned with the internal wave at the interface of two layers of liquids due to a hydrofoil in the lower layer liquid. The two-layer fluid is assumed moving parallel to the interface at different velocities. The stratified flow is modeled based on the incompressible potential flow theory, with the nonlinear boundary conditions at the interface. Boundary integral equations are formulated for the fully nonlinear interfacial wave generated by the hydrofoil. The numerical model results in a set of nonlinear algebra equations, which are solved using the quasi-Newton method. We show that the quasi-Newton method is more efficient than Newton’s method, which is often used for solving these types of equations in the literature. The wave profiles were analyzed in terms of the location and thickness of the hydrofoil, the Froude number, and the ratio of the densities of the two fluids. The computations show that the interfacial wave amplitude showed a trend first of increase and then of decrease with the distance between the hydrofoil and the still interface.

Study on the interactions between two identical oscillation bubbles and a free surface in a tank

A boundary element method based on the incompressible potential flow theory is adopted to investigate the interaction between two identical oscillating bubbles and a free surface in a tank. An axisymmetric numerical model is established, and certain numerical techniques are proposed to address coefficient matrix singularity and fluid-structure intersection. Experiments with spark-generated bubbles in a cylindrical tank recorded by a high-speed camera are conducted, and the numerical results are validated. On this basis, a typical case of bubbles interacting with a free surface in a tank with relatively small inter-bubble and bubble-free surface distances is carefully studied. A crown-shaped water column at the free surface is observed both numerically and experimentally. The maximum volume of the lower bubble is found to be much larger than that of the upper one. The effects of the inter-bubble and bubble-wall distances on bubble dynamics and free surface motion are analyzed. The results can provide a useful reference for underwater explosion experiments in the confined fluid field.

An approximate analytical method to size an air vessel in a water supply system

The size parameters of an air vessel, mainly including its initial gas volume and total volume, account for its performance in hydraulic transients. Traditionally, the size of an air vessel in a given water supply system is estimated by trial and error, usually aided by computers. To simplify the preliminary estimation of the air vessel's volume, in this paper, an approximate analytical method is proposed and derived on the basis of incompressible flow theory and simple harmonic vibration theory. With this method, a theoretically optimal size of an air vessel, which is able to meet the protective requirements in hydraulic transients, could easily be obtained in an analytical way. Besides, a real engineering case is introduced in this paper to illustrate that the proposed method could give guidance and reference in the preliminary work of designing an air vessel in practice.