Viscous Dissipation Rate Research Articles

The cost of swimming in generalized Newtonian fluids: experiments withC. elegans

Numerous natural processes are contingent on microorganisms’ ability to swim through fluids with non-Newtonian rheology. Here, we use the model organismCaenorhabditis elegansand tracking methods to experimentally investigate the dynamics of undulatory swimming in shear-thinning fluids. Theory and simulation have proposed that the cost of swimming, or mechanical power, should be lower in a shear-thinning fluid compared to a Newtonian fluid of the same zero-shear viscosity. We aim to provide an experimental investigation into the cost of swimming in a shear-thinning fluid from (i) an estimate of the mechanical power of the swimmer and (ii) the viscous dissipation rate of the flow field, which should yield equivalent results for a self-propelled low Reynolds number swimmer. We find the cost of swimming in shear-thinning fluids is less than or equal to the cost of swimming in Newtonian fluids of the same zero-shear viscosity; furthermore, the cost of swimming in shear-thinning fluids scales with a fluid’s effective viscosity and can be predicted using fluid rheology and simple swimming kinematics. Our results agree reasonably well with previous theoretical predictions and provide a framework for understanding the cost of swimming in generalized Newtonian fluids.

Numerical investigation of interaction between rising bubbles in a viscous liquid

The rising behavior of bubbles undergoing bubble-bubble interaction in a viscous liquid is studied using a two-dimensional direct numerical simulation. Level contour reconstruction method (LCRM), one of the connectivity-free front tracking methods, is applied to describe a moving interface accurately under highly deformable conditions. This work focuses on the effects of bubble size on the interaction of two bubbles rising side-by-side in a stagnant liquid. Several characteristics of bubble-bubble interaction are analyzed quantitatively as supported by energy analysis. The results showed clear differences between small and large bubbles with respect to their interaction behavior in terms of lateral movement, vortex intensity, suppression of surface deformation, and viscous dissipation rate. Distributions of vorticity and viscous dissipation rate near the bubble interfaces also differed depending on the size of the bubbles. Strong vortices from large bubbles triggered oscillation in bubble-bubble interaction and played a dominant role in the interaction process as the size of bubbles increases.

Scaling law for bubbles rising near vertical walls

This paper examines the rising motion of a layer of gas bubbles next to a vertical wall in a liquid in the presence of an upward flow parallel to the wall to help with the understanding of the fluid dynamics in a bubbly upflow in vertical channels. Only the region near the wall is simulated with an average pressure gradient applied to the domain that balances the weight of the liquid phase. The upward flow is created by the rising motion of the bubbles. The bubbles are kept near the wall by the lateral lift force acting on them as a result of rising in the shear layer near the wall. The rise velocity of the bubbles sliding on the wall and the average rise velocity of the liquid depend on three dimensionless parameters, Archimedes number, Ar, Eötvös number, Eo, and the average volume fraction of bubbles on the wall. In the limit of small Eo, bubbles are nearly spherical and the dependency on Eo becomes negligible. In this limit, the scaling of the liquid Reynolds number with Archimedes number and the void fraction is presented. A scaling argument is presented based on viscous dissipation analysis that matches the numerical findings. Viscous dissipation rates are found to be high in a thin film region between the bubble and the wall. A scaling of the viscous dissipation and steady state film thickness between the bubble and the wall with Archimedes number is presented.

Are there two regimes in strongly rotating turbulence?

We describe numerical experiments of freely decaying, rapidly rotating turbulence in which the Rossby number varies from Ro = O(1) down to Ro ∼ 0.02. Our central premise is that there exists two distinct dynamical regimes; one for Ro > 0.3 → 0.4, which is typical of most laboratory experiments, and another corresponding to Ro < 0.3, which covers most previous numerical studies. The case of Ro > 0.3 → 0.4 is reported in Baqui and Davidson [“A phenomenological theory of rotating turbulence,” Phys. Fluids 27, 025107 (2015)] and is characterised by: (i) a growth of the parallel integral scale according to l|| ∼ l⊥Ωt; (ii) a dissipation law which is quite different from that predicted by weak-turbulence theories, specifically ε = βu3/l|| where the pre-factor β is a constant of order unity; and (iii) an inertial-range energy spectrum for both the parallel and perpendicular wavenumbers which scales as k−5/3, a scaling that has nothing to do with Kolmogorov’s law in non-rotating turbulence. (Here, l|| is the integral length-scale parallel to the rotation vector Ω, l⊥ the integral length-scale perpendicular to Ω, u the integral scale velocity, and ε the viscous dissipation rate per unit mass.) By contrast, in the low-Ro regime, we find that l|| ∼ l⊥Ωt is replaced by l|| ∼ ut and there is no power-law scaling of the inertial range energy spectrum. While the dissipation law ε = βu3/l|| continues to hold at low Ro, at least approximately, the value of β now depends on Ro. It appears, therefore, that the dynamics of these two regimes are very different, and this may help explain why experimentalists and theoreticians sometimes present rather different interpretations of rotating turbulence.

Energy balance in viscous liquid containing a bubble: Rise due to buoyancy

The total energy balance of a system consisting of a bubble (radius 0.74 mm) freely rising in viscous liquid under the action of buoyancy was considered. Values of bubble acceleration, local and terminal velocities, and shape deformations calculated by numerically solving the Navier‐Stokes equation are compared with experimentally determined values. Additionally, the energies of the system associated with bubble motion (kinetic, potential, and rate of viscous energy dissipation), obtained from simulations, are given. On the basis of energy balances obtained for the system, two independent relations describing the motion of a bubble (Lagrange and Joseph equations) were used to show that, for steady‐state conditions, the constant (terminal) velocity of the bubble is a result of balance between buoyancy force and the drag originating from energy dissipation due to fluid viscosity. According to the calculations performed, the discrepancy between the expected and actual results of the Lagrange and Joseph equations for the system considered was ∼1 %. The estimated added mass coefficient was in close agreement with expected theoretical value for deformed spheres, and increased with decreasing tube radius.

Dynamics of cracking in drying colloidal sheets.

Colloidal dispersions are known to display a fascinating network of cracks on drying. We probe the fracture mechanics of free-standing films of aqueous polymer-particle dispersions. Thin films of the dispersion are cast between a pair of plain steel wires and allowed to dry under ambient conditions. The strain induced on the particle network during drying is relieved by cracking. The stress which causes the films to crack has been calculated by measuring the deflection of the wires. The critical cracking stress varied inversely to the two-thirds' power of the film thickness. We also measure the velocity of the tip of a moving crack. The motion of a crack has been modeled as a competition between the release of the elastic energy stored in the particle network, the increase in surface energy as a result of the growth of a crack, the rate of viscous dissipation of the interstitial fluid and the kinetic energy associated with a moving crack. There is fair agreement between the measured crack velocities and predictions.

A generalized model for optimal transport of images including dissipation and density modulation

In this paper the optimal transport and the metamorphosis perspectives are combined. For a pair of given input images geodesic paths in the space of images are defined as minimizers of a resulting path energy. To this end, the underlying Riemannian metric measures the rate of transport cost and the rate of viscous dissipation. Furthermore, the model is capable to deal with strongly varying image contrast and explicitly allows for sources and sinks in the transport equations which are incorporated in the metric related to the metamorphosis approach by Trouve and Younes. In the non-viscous case with source term existence of geodesic paths is proven in the space of measures. The proposed model is explored on the range from merely optimal transport to strongly dissipative dynamics. For this model a robust and effective variational time discretization of geodesic paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals. These functionals are defined on corresponding pairs of intensity functions and on associated pairwise matching deformations. Existence of time discrete geodesics is demonstrated. Furthermore, a finite element implementation is proposed and applied to instructive test cases and to real images. In the non-viscous case this is compared to the algorithm proposed by Benamou and Brenier including a discretization of the source term. Finally, the model is generalized to define discrete weighted barycentres with applications to textures and objects.

Turbulent Rayleigh–Bénard convection in spherical shells

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a Prandtl number of unity and for Rayleigh numbers up to $10^{9}$. In this geometry, curvature and radial variations of the gravitational acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from ${\it\eta}=r_{i}/r_{o}=0.2$ to ${\it\eta}=0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl–Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with ${\it\eta}=0.6$ and with gravity proportional to $1/r^{2}$. We show that our spherical models are consistent with the predictions of Grossmann & Lohse’s (J. Fluid Mech., vol. 407, 2000, pp. 27–56) theory and that $\mathit{Nu}(\mathit{Ra})$ and $\mathit{Re}(\mathit{Ra})$ scalings are in good agreement with plane layer results.

A novel design method based on flow pattern construction for flow passage with low flow drag and pressure drop

A novel design method for flow passage with low flow drag and pressure drop was developed based on flow pattern construction. The specific flow pattern with minimum viscous dissipation rate represents low pressure drop in the flow passage. Thus, the fluid flow requires to be optimized as a variational problem subject to the minimum viscous dissipation rate principle, which is also considered as flow pattern construction. By using calculus of variations, the governing equations for the construction of the specific flow pattern were derived from the functional extremum problem represented by the Lagrange multiplier equation. After that, a computational fluid dynamics model was set up and applied in the solution procedure of the governing equations to generate the geometry of the flow passages based on the flow pattern construction. As the illustrative examples to validate the proposed design method, the geometries of flow passages were optimized in terms of pressure drop in a wide range of Re numbers.

Effect of trabeculae and papillary muscles on the hemodynamics of the left ventricle

The impact of surface trabeculae and papillary muscles on the hemodynamics of the left ventricle (LV) is investigated using numerical simulations. Simulations of ventricular flow are conducted for two different models of the LV derived from high-resolution cardiac computed tomography (CT) scans using an immersed boundary method-based flow solver. One model comprises a trabeculated left ventricle (TLV) that includes both trabeculae and papillary muscles, while the second model has a smooth left ventricle that is devoid of any of these surface features. Results indicate that the trabeculae and papillary muscles significantly disrupt the vortices that develop during early filling in the TLV model. Large recirculation zones are found to form in the wake of the papillary muscles; these zones enhance the blockage provided by the papillary muscles and create a path for the mitral jet to penetrate deeper into the ventricular apex during diastole. During systole, the trabeculae enhance the apical washout by ‘squeezing’ the flow from the apical region. Finally, the trabeculae enhance viscous dissipation rate of the ventricular flow, but this effect is not significant in the overall power budget.

Wave Boundary Layer Turbulence over Surface Waves in a Strongly Forced Condition

AbstractAccurate predictions of the sea state–dependent air–sea momentum flux require a thorough understanding of the wave boundary layer turbulence over surface waves. A set of momentum and energy equations is derived to formulate and analyze wave boundary layer turbulence. The equations are written in wave-following coordinates, and all variables are decomposed into horizontal mean, wave fluctuation, and turbulent fluctuation. The formulation defines the wave-induced stress as a sum of the wave fluctuation stress (because of the fluctuating velocity components) and a pressure stress (pressure acting on a tilted surface). The formulations can be constructed with different choices of mapping. Next, a large-eddy simulation result for wind over a sinusoidal wave train under a strongly forced condition is analyzed using the proposed formulation. The result clarifies how surface waves increase the effective roughness length and the drag coefficient. Specifically, the enhanced wave-induced stress close to the water surface reduces the turbulent stress (satisfying the momentum budget). The reduced turbulent stress is correlated with the reduced viscous dissipation rate of the turbulent kinetic energy. The latter is balanced by the reduced mean wind shear (satisfying the energy budget), which causes the equivalent surface roughness to increase. Interestingly, there is a small region farther above where the turbulent stress, dissipation rate, and mean wind shear are all enhanced. The observed strong correlation between the turbulent stress and the dissipation rate suggests that existing turbulence closure models that parameterize the latter based on the former are reasonably accurate.

Water-Channel Study of Flow and Turbulence Past a Two-Dimensional Array of Obstacles

A neutral boundary layer was generated in the laboratory to analyze the mean velocity field and the turbulence field within and above an array of two-dimensional obstacles simulating an urban canopy. Different geometrical configurations were considered in order to investigate the main characteristics of the flow as a function of the aspect ratio (AR) of the canopy. To this end, a summary of the two-dimensional fields of the fundamental turbulence parameters is given for AR ranging from 1 to 2. The results show that the flow field depends strongly on AR only within the canyon, while the outer flow seems to be less sensitive to this parameter. This is not true for the vertical momentum flux, which is one of the parameters most affected by AR, both within and outside the canyon. The experiments also indicate that, when (i.e. the skimming flow regime), the roughness sub-layer extends up to a height equal to 1.25 times the height of the obstacles (H), surmounted by an inertial sub-layer that extends up to 2.7 H. In contrast, for (i.e. the wake-interference regime) the inertial sub-layer is not present. This has significant implications when using similarity laws for deriving wind and turbulence profiles in canopy flows. Furthermore, two estimations of the viscous dissipation rate of turbulent kinetic energy of the flow are given. The first one is based on the fluctuating strain rate tensor, while the second is related to the mean strain rate tensor. It is shown that the two expressions give similar results, but the former is more complicated, suggesting that the latter might be used in numerical models with a certain degree of reliability. Finally, the data presented can also be used as a dataset for the validation of numerical models.

Effective surface dilatational viscosity of highly concentrated particle-laden interfaces.

The effective surface dilatational viscosity is calculated of a flat interface separating two immiscible fluids laden with half-immersed monodisperse rigid spherical non-Brownian particles in the limit of high particle concentration. The derivation is based upon the facts that (i) highly concentrated particle arrays in a plane form a hexagonal structure and (ii) the dominant contribution to the viscous dissipation rate arises in the thin gaps between neighboring particles.

Numerical Comparison of Rushton Turbine and CD-6 Impeller in Non-Newtonian Fluid Stirred Tank

A computational fluid dynamics simulation is done for non-Newtonian fluid in a baffled stirred tank. The CMC solution is taken as non-Newtonian shear thinning fluid for simulation. The Reynolds Average Navier Stocks equation with steady state multi reference frame approach is used to simulate flow in the stirred tank. The turbulent flow field is modelled using realizable k-e turbulence model. The simulated velocity profiles of Rushton turbine is validated with literature data. Then, the simulated flow field of CD-6 impeller is compared with the Rushton turbine. The flow field generated by CD-6 impeller is less in magnitude than the Rushton turbine. The impeller global parameter, power number and flow number, and entropy generation due to viscous dissipation rate is also reported. Keywords—Computational fluid dynamics, non-Newtonian, Rushton turbine, CD-6 impeller, power number, flow number, viscous dissipation rate.

Extending the bounds of ‘steady’ RANS closures: Toward an instability-sensitive Reynolds stress model

The incapability of the conventional Unsteady RANS (Reynolds–Averaged Navier Stokes) models to adequately capture turbulence unsteadiness presents the prime motivation of the present work, which focuses on formulating an instability-sensitive, eddy-resolving turbulence model on the Second-Moment Closure level. The model scheme adopted, functioning as a ‘sub-scale’ model in the Unsteady RANS framework, represents a differential near-wall Reynolds stress model formulated in conjunction with the scale-supplying equation governing the homogeneous part of the inverse turbulent time scale ωh (ωh=ɛh/k). The latter equation was straightforwardly obtained from the model equation describing the dynamics of the homogeneous part of the total viscous dissipation rate ɛ, defined as ɛh=ɛ−0.5ν∂2k/(∂xj∂xj) (Jakirlic and Hanjalic, 2002), by applying the derivation rules to the expression for ωh. The model capability to account for vortex length and time scales variability was enabled through an additional term in the corresponding length-scale determining equation, providing a selective enhancement of its production, pertinent particularly to the highly unsteady separated shear layer region, modeled in terms of the von Karman length scale (comprising the second derivative of the velocity field) in line with the SAS (Scale-Adaptive Simulation) proposal (Menter and Egorov, 2010). The present model formulation, termed as SRANS model (Sensitized RANS), does not comprise any parameter depending explicitly on grid spacing. The predictive capabilities of the newly proposed length-scale determining model equation, solved in conjunction with Jakirlic and Hanjalic’s (2002) Reynolds stress model equation, are presently demonstrated by computing the flow configurations of increasing complexity featured by boundary layer separation from sharp-edged and continuous curved surfaces: backward-facing step flow, flow over a wall-mounted fence, flow over smoothly contoured periodically arranged hills and flow in a 3-D diffuser. The model performances are also assessed in capturing the natural decay of the homogeneous isotropic turbulence and the near-wall Reynolds stress anisotropy in a plane channel. In most cases considered the fluctuating velocity field was obtained starting from steady RANS results.

Full and Partial Emulsification of Crude Oil–Water Systems as a Function of Shear Intensity, Water Fraction, and Temperature

Emulsification of crude oil and water may significantly affect the flow characteristics of the multiphase flow in gathering lines. Obviously, emulsification of a crude oil–water system is greatly affected by conditions such as shear intensity, water fraction, and temperature. However, there has been a lack of quantitative study of the emulsified water fraction of crude oil–water systems under various conditions of water fraction, temperature, and shear intensity. In this work, experiments were conducted by using a stirred vessel calibrated for the determination of mean shear rate, and the emulsified water fractions under flowing conditions were determined by extrapolation of the separated water volume to the moment when stirring was just terminated. Full emulsification of water was observed in a specific range of shear rates and below a critical water fraction, and only partial emulsification might occur above this critical water fraction. The range of shear rates in which full emulsification might occur became larger with a decrease in the water fraction, but it did not change with temperature in the studied range from 40 to 90 °C. For partial emulsification, the emulsified water fraction under flowing conditions was found to correlate well with the entropy production rate of viscous flow or energy dissipation rate regardless of the differences in water fraction, temperature, and shear rate, and a power law equation might well fit the data.

Hydrodynamic Trails Produced by Daphnia: Size and Energetics

This study focuses on quantifying hydrodynamic trails produced by freely swimming zooplankton. We combined volumetric tracking of swimming trajectories with planar observations of the flow field induced by Daphnia of different size and swimming in different patterns. Spatial extension of the planar flow field along the trajectories was used to interrogate the dimensions (length and volume) and energetics (dissipation rate of kinetic energy and total dissipated power) of the trails. Our findings demonstrate that neither swimming pattern nor size of the organisms affect the trail width or the dissipation rate. However, we found that the trail volume increases with increasing organism size and swimming velocity, more precisely the trail volume is proportional to the third power of Reynolds number. This increase furthermore results in significantly enhanced total dissipated power at higher Reynolds number. The biggest trail volume observed corresponds to about 500 times the body volume of the largest daphnids. Trail-averaged viscous dissipation rate of the swimming daphnids vary in the range of to and the observed magnitudes of total dissipated power between and , respectively. Among other zooplankton species, daphnids display the highest total dissipated power in their trails. These findings are discussed in the context of fluid mixing and transport by organisms swimming at intermediate Reynolds numbers.

植栽を有する流れ場のLESモデルの提案

In the previous paper, I proposed a one-equation-type subgrid-scale (SGS) model for a Large Eddy Simulation (LES) of plant canopy flows. In that paper, the SGS turbulent energy equation was derived from the equations of continuity and Navier-Stokes. The SGS turbulent energy equation was closed by modeling several unknown terms which appeared in the equation. In this paper I tried to modify the modeling of the turbulent diffusion terms and the viscous dissipation rate. The turbulent diffusion terms were modeled in a unified way. The viscous dissipation rate was modeled using two length scales.

Global existence of smooth solutions to the $k$-$\epsilon$-model equations for turbulent flows

In this paper we are concerned with the global existence of smooth solutions to the k-epsilon model equations for turbulent flows in R-3. The global well-posedness is proved under the condition that the initial data are close to the standard equilibrium state in the H-3-framework. The proof relies on energy estimates on velocity, temperature, turbulent kinetic energy, and the rate of viscous dissipation. We use several new techniques to overcome the difficulties from the product of two functions and higher order norms. This is the first result concerning k-epsilon model equations.

A model for the viscous dissipation rate in stably stratified, sheared turbulence

A model for the turbulence dissipation rate in stably stratified shear turbulence is developed and validated. The functional dependence of the model is derived from first principles and it represents a conceptually new approach in that it depends on the background temperature field rather than on the fluctuating velocity field. This novel feature makes the proposed model a viable candidate for dissipation rate estimates in measured real‐life flows. Direct numerical simulation data are used for a priori assessment of the proposed model. It is demonstrated that the proposed model performs very well, particularly in cases where the background stratification becomes dynamically important. Also, a generalized expression for the mixing coefficient has been rigorously derived from first principles assuming local isotropy of incompressible turbulent flows. The mixing coefficient is shown to depend on the Prandtl number and values are in correspondence with previous studies.