Viscous Extension Research Articles

Phase-field simulations of droplet impact on superhydrophobic surfaces

Although the impingement of droplets on superhydrophobic surfaces has been extensively studied, the critical role of the air is typically ignored, leading to some paradoxical conclusions. In this paper, a phase-field model with dynamic contact angles is implemented to simulate the impingement of droplets on a superhydrophobic surface. The simulation results agree well with the experimental data. The droplet's impact on the surface is observed and divided into four categories: contactless bouncing, wet bouncing, dry-out bouncing and non-bouncing. The role of air was emphasised by analysing pressure distribution and velocity field corresponding to the droplet evolution, revealing the mechanism behind the droplet impact. A non-monotonically varying trend of the non-dimensional maximum wetting diameter is observed by increasing Ohnesorge number (Oh). The value of dimensionless viscous extension first increased and then decreased with increasing impact number, and this counter-intuitively outcome has not been reported previously. Non-dimensional contact time of a bouncing droplet decreases rapidly with increasing Weber number (We), but approaches a plateau value if We is greater than 1. Additionally, for liquids with different viscosities, the non-dimensional contact time of bouncing droplets is independent of Oh. Moreover, regime maps of the dynamic behaviours of a droplet impact are developed to present an intuitive interpretation of different droplet fates.

State-Space Modeling of Viscous Unsteady Aerodynamic Loads

In this paper, we summarize our recently developed viscous unsteady theory, which couples potential flow with the triple-deck boundary-layer theory. This approach provides a viscous extension of potential-flow unsteady aerodynamics. As such, a Reynolds-number-dependent transfer function is determined for unsteady lift. We then use the Wiener–Hammerstein structure to develop a finite-dimensional approximation of such an infinite-dimensional theory, presenting it in a state-space model. This novel nonlinear state-space model of viscous unsteady aerodynamic loads is expected to serve aerodynamicists better than the classical Theodorsen’s model because it captures viscous effects (that is, Reynolds number dependence) as well as nonlinearity and additional lag in the lift dynamics; it also allows simulation of arbitrary time-varying airfoil motions (not necessarily harmonic). Moreover, being in a state-space form makes it quite convenient for simulation and coupling with structural dynamics to perform aeroelasticity, flight dynamics analysis, and control design. We then develop a linearization of such a model, which enables analytical results. Subsequently, we derive an analytical representation of the viscous lift frequency response function: an explicit function of both the frequency and Reynolds number. We also develop a state-space model of the linearized response. We finally simulate the nonlinear and linear models to a nonharmonic small-amplitude pitching maneuver at a Reynolds number of 100,000 and compare the resulting lift and pitching moment with those obtained from potential flow; this is in reference to relatively higher-fidelity computations of the unsteady Reynolds-averaged Navier–Stokes equations.

Viscous extension of vortex methods for unsteady aerodynamics

The Kutta condition has been extensively used to determine aerodynamic loads of steady and unsteady flows over airfoils. Nevertheless, the application of this condition to unsteady flows has been controversial for decades. A viscous correction to the Kutta condition was recently developed by matching the potential flow solution with a special boundary layer theory that resolves the flow field in the immediate vicinity of the trailing edge: the triple-deck boundary layer theory. In this work, we utilize this viscous condition to extend two common numerical methods for unsteady aerodynamics to capture viscous effects on the dynamics of unsteady lift and pitching moment—we develop viscous versions of the traditional discrete vortex method and unsteady vortex lattice/panel method. The resulting aerodynamic loads obtained from the proposed numerical models are compared against higher-fidelity simulations of unsteady Reynolds-averaged Navier–Stokes equations for an airfoil undergoing step, harmonic, and complex maneuvers. The obtained results are consistently in better agreement with the unsteady Reynolds-averaged Navier–Stokes simulations in comparison to their potential flow counterpart. In conclusion, the developed numerical methods are capable of capturing (i) unsteady effects; (ii) viscous effects (e.g., viscosity-induced lag) on the dynamics of lift and moment at high frequencies and low Reynolds numbers; and (iii) wake deformation, for arbitrary time-varying motion.

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

Predicting Rotor Heat Transfer Using the Viscous Blade Element Momentum Theory and Unsteady Vortex Lattice Method

Calculating the unsteady convective heat transfer on helicopter blades is the first step in the prediction of ice accretion and the design of ice-protection systems. Simulations using Computational Fluid Dynamics (CFD) successfully model the complex aerodynamics of rotors as well as the heat transfer on blade surfaces, but for a conceptual design, faster calculation methods may be favorable. In the recent literature, classical methods such as the blade element momentum theory (BEMT) and the unsteady vortex lattice method (UVLM) were used to produce higher fidelity aerodynamic results by coupling them to viscous CFD databases. The novelty of this research originates from the introduction of an added layer of the coupling technique to predict rotor blade heat transfer using the BEMT and UVLM. The new approach implements the viscous coupling of the two methods from one hand and introduces a link to a new airfoil CFD-determined heat transfer correlation. This way, the convective heat transfer on ice-clean rotor blades is estimated while benefiting from the viscous extension of the BEMT and UVLM. The CFD heat transfer prediction is verified using existing correlations for a flat plate test case. Thrust predictions by the implemented UVLM and BEMT agree within 2% and 80% compared to experimental data. Tip vortex locations by the UVLM are predicted within 90% but fail in extreme ground effect. The end results present as an estimate of the heat transfer for a typical lightweight helicopter tail rotor for four test cases in hover, ground effect, axial, and forward flight.

Improvement of viscous substance production during Cheonggukjang fermentation added with glycine.

When Bacillus subtilis NB-NUC1 associated with Cheonggukjang fermentation was aerobically grown in a synthetic medium containing 1 to 2% glycine (w/v), cell growth was inhibited in a dose-dependent manner. Subsequently, different concentrations of glycine (0, 1, and 2%) were used in Cheonggukjang fermentation for 96h at 40°C. Supplementation of 1% glycine increased extracellular γ-glutamyl transpeptidase (γ-GTPase), responsible for the production of viscous substance. Based on correlation studies, we conclude that the production of viscous substance is correlated with viscous extension (r = 0.867), extracellular proteins contents (r = 0.821), and γ-GTPase activity (r = 0.807). The molecular weight of the viscous substance obtained during Cheonggukjang fermentation by B. subtilis NB-NUC1 was also affected by glycine supplementation. Our results demonstrate that glycine supplementation before solid-state fermentation may increase the mass production of mucilage in food industry.

Extension of classical stability theory to viscous planar wall-bounded shear flows

A viscous extension of Arnold’s inviscid theory for planar parallel non-inflectional shear flows is developed and a viscous Arnold’s identity is obtained. Special forms of the viscous Arnold’s identity have been revealed that are closely related to the perturbation’s enstrophy identity derived by Synge (Proceedings of the Fifth International Congress for Applied Mechanics, 1938, pp. 326–332, John Wiley) (see also Fraternaleet al.,Phys. Rev. E, vol. 97, 2018, 063102). Firstly, an alternative derivation of the perturbation’s enstrophy identity for strictly parallel shear flows is acquired based on the viscous Arnold’s identity. The alternative derivation induces a weight function. Thereby, a novel weighted perturbation’s enstrophy identity is established, which extends the previously known enstrophy identity to include general streamwise translation-invariant shear flows. Finally, the validity of the enstrophy identity for parallel shear flows is rigorously examined and established under global nonlinear dynamics imposed with two classes of wall boundary conditions. As an application of the enstrophy identity, we quantitatively investigate the mechanism of linear instability/stability within the normal modal framework. The investigation reveals a subtle interaction between a critical layer and its adjacent boundary layer, which determines the stability nature of the disturbance. As an implementation of the relaxed wall boundary conditions imposed for the enstrophy identity, a control scheme is proposed that transitions the wall settings from the no-slip condition to the free-slip condition, through which a flow is stabilized quickly in an early stage of the transition.

Viscous extension of potential-flow unsteady aerodynamics: the lift frequency response problem

The application of the Kutta condition to unsteady flows has been controversial over the years, with increased research activities over the 1970s and 1980s. This dissatisfaction with the Kutta condition has been recently rejuvenated with the increased interest in low-Reynolds-number, high-frequency bio-inspired flight. However, there is no convincing alternative to the Kutta condition, even though it is not mathematically derived. Realizing that the lift generation and vorticity production are essentially viscous processes, we provide a viscous extension of the classical theory of unsteady aerodynamics by relaxing the Kutta condition. We introduce a trailing-edge singularity term in the pressure distribution and determine its strength by using the triple-deck viscous boundary layer theory. Based on the extended theory, we develop (for the first time) a theoretical viscous (Reynolds-number-dependent) extension of the Theodorsen lift frequency response function. It is found that viscosity induces more phase lag to the Theodorsen function particularly at high frequencies and low Reynolds numbers. The obtained theoretical results are validated against numerical laminar simulations of Navier–Stokes equations over a sinusoidally pitching NACA 0012 at low Reynolds numbers and using Reynolds-averaged Navier–Stokes equations at relatively high Reynolds numbers. The physics behind the observed viscosity-induced lag is discussed in relation to wake viscous damping, circulation development and the Kutta condition. Also, the viscous contribution to the lift is shown to significantly decrease the virtual mass, particularly at high frequencies and Reynolds numbers.

Flexible Multibody Systems with Large Deformations and Nonlinear Structural Damping Using Absolute Nodal Coordinates

A formulation for flexible Multibody Systems (MBS) is proposed where flexible bodies are described using absolute coordinates forisoparametric hexagonal elements under consideration of nonlinearstructural damping.The use of absolute coordinates together with a nonlinear finite-element formulation allows for large deformations andprovides an accurate description of rigid bodymotion and inertia in the case of large rotations without additionalconsiderations. Further, constant mass matrices are obtained for isoparametric elements. Hexagonal elements are important, e.g., if general solid bodies with low stiffness, i.e.not negligible large deformations, are part of the MBS and cannot be modeled usingbeam, plate, or shell elements. Since only nodal translational degrees of freedom are used for hexagonal elements,additional questions arise. For example, imposing jointconstraints for relative rotations between two bodies requiresa nodal reference frame at connection points. Flexible bodies underlie structural damping. An approach is proposed in order to consider such damping using a viscous extension for hyper-elastic materials of neo-Hooke type.

Dripping into subterranean cavities from unsaturated fractures under evaporative conditions

Water dripping into subterranean cavities within fractured porous media is studied in order to improve estimates of dripping rates, drop sizes, and chemical composition of droplets that could affect long‐term integrity of waste disposal canisters placed in caverns. Steady state liquid flux in fracture surfaces supported by flow in partially liquid‐filled grooves and liquid films in adjacent planes was calculated as a function of the matric potential (vapor pressure) of the fracture. At an intersection of a vertical fracture with a wider cavity the liquid flux feeds a growing pendant drop that eventually detaches. Equilibrium state size and approximate shape of liquid drops suspended from the cavity ceiling were determined from lateral and vertical force balance considering capillarity, gravity, and hydrostatic pressure. A one‐dimensional, viscous extension model with appropriate gravitational and surface tension components was employed to determine dripping rate from specified fracture roughness geometry as a function of matric potential (flux). The effect of evaporation from drop surface during drop formation was incorporated; the resulting alterations in drop volume, dripping rate, and drop solute concentration were determined. To facilitate experimental testing of the proposed model, a decoupled solution that considers independently controlled flux and evaporation is presented. Under evaporative conditions, dripping in finite period is possible only when volumetric flux exceeds evaporative demand. Calculations indicate that dripping rate and solute concentration are extremely sensitive to ambient matric potential. The results of this work may be extended to study other phenomena including formation and growth of stalactites and rivulet flow in cave ceilings.

Power Requirements for Large-Amplitude Flapping Flight

In this paper, a method is presented for computing the circulation distribution along the span of a e apping wing that minimizes the power required to generate a prescribed lift and thrust. The power is composed of three parts: useful thrust power, induced power, and proe le power. Here, the thrust and induced power are expressed in terms of the Kelvin impulse and kinetic energy associated with the sheet of trailing and shed vorticity left behind the e apping wing. The proe le power is computed using a quasisteady approximation of the two-dimensional viscous drag polar at each spanwise station of the wing. A variational principle is then formed to determine the necessary conditions for the circulation distribution to be optimal. Included in the variational principle is a constraint that the wing not stall. This variational principle, which is essentially the viscous extension of the well-known Betz criterion for optimal propellers, is discretized using a vortex-lattice model of the wake, and the optimum solution is computed numerically. The present method is used to analyze a conventional propeller as well as a rigid wing in forward-e ight e apping about a hinge point on the longitudinal axis.

Rheology of Layered Thermoplastic Matrix Composites during Compression Molding

Abstract In view of optimizing the industrial compression molding process of thermoplastic composites, the rheological and microstructural behavior of a polypropylene/glass fiber composite is investigated in model squeeze-flow geometries. The overall stress / strain behavior of the material at various compression rates is recorded and the induced orientation of the fibers is investigated by means of a special electron microscopic characterization method. By contrast to pure polypropylene, it is shown that in the high speed range, the macroscopic flow process is controlled by both the viscous extension and the relative sliding of parallel fibrous layers, the latter becoming unstable when the flow undergoes a rapid transition from divergent to convergent. Under high pressure, voids are dissolved in the polymer melt. In the case of non-isothermal compression, the rheology of the composite is not significantly affected by the cooling of the surface layers. The multilayer plug-flow model based on the sliding mechanism of viscous layers is found to reproduce correctly the experimental stress/strain behavior.

Multilayer plug flow modeling of the fast stamping process for a polypropylene/glass fiber composite

AbstractThe isothermal compression at 200°C of a polypropylene/glass fiber composite was investigated in both axisymmetric and unidirectional flows. The overall stress/strain behavior at various compression strain rates was recorded and the induced orientation of the fibers is investigated. It is shown that, in the high speed range, the macroscopic flow process controlled by both the viscous extension and the relative sliding of parallel fibrous layers. The evolution of the compression force is modeled as the sum of a viscous resistance and an interlaminar friction.

Detection of changes in modulus and viscosity of wheat flour doughs by the “work technique” of muller et al

AbstractAt a given total extension both the recoverable (elastic) and the irrecoverable (viscous) work, as determined by Muller et al.3 with the Brabender extensograph, depend on the modulus as well as on the viscosity of the test‐piece. A theoretical analysis shows that, at constant elastic extension, the elastic work depends on the modulus only, and is not affected by changes in viscosity. On the other hand, if a few simplifying assumptions are permitted, the total work at constant viscous extension depends on the viscosity, but not on the modulus. Therefore, plots of elastic work against (a function of) the elastic extension and of the total work against the viscous extension permit easier detection of changes in modulus and viscosity than do plots of elastic and viscous work against the total extension.

A Coefficient of Viscous Extension

WHILE working on a generalized theory on straining of amorphous or quasi-amorphous solids, it became clear that one could include the straining of viscous fluids by means of a suitable stress-strain parameter λ which I call a ‘coefficient of viscous extension' related to the ‘coefficient of viscous shear' usually called1 the coefficient of viscosity µ. In this communication only the flow strain component of a deformed substance2 is considered, so that the elastic and plastic components3 are neglected and, further, only isotropic substances are discussed.