Viscous Stress Terms Research Articles

Green's function integral method for pressure reconstruction from measured pressure gradient and the interpretation of omnidirectional integration

Accurately and efficiently measuring the pressure field is of paramount importance in many fluid mechanics applications. The pressure gradient field of a fluid flow can be determined from the balance of the momentum equation based on the particle image velocimetry measurement of the flow kinematics, which renders the experimental evaluation of the material acceleration and the viscous stress terms possible. In this paper, we present a novel method of reconstructing the instantaneous pressure field from the error-embedded pressure gradient measurement data. This method utilized the Green's function of the Laplacian operator as the convolution kernel that relates pressure to the pressure gradient. A compatibility condition on the boundary offers equations to solve for the boundary pressure. This Green's function integral (GFI) method has a deep mathematical connection with the state-of-the-art omnidirectional integration (ODI) for pressure reconstruction. As mathematically equivalent to ODI in the limit of an infinite number of line integral paths, GFI spares the necessity of line integration along zigzag integral paths, rendering generalized implementation schemes for both two and three-dimensional problems with arbitrary inner and outer boundary geometries while bringing in improved computational simplicity. In the current work, GFI is applied to pressure reconstruction of simple canonical and isotropic turbulence flows embedded with error in two-dimensional and three-dimensional domains, respectively. Uncertainty quantification is performed by eigenanalysis of the GFI operator in domains with both simply and multiply connected shapes. The accuracy and the computational efficiency of GFI are evaluated and compared with ODI.

Rarefaction effect on the aerodynamics of bristled wings in miniature insects

All previous studies on the aerodynamics of bristled wings in miniature insects are based on continuum flows. However, the diameter of the bristle is very small, and the diameter-based Knudsen number (Kn) is approximately between 0.03 and 0.11, indicating that the flow around the bristle is in the slip-flow regime and rarefaction effect will be present. To investigate how the rarefaction will affect the aerodynamic force and flow field of the bristled wing, we calculated and analyzed the flow around a model bristled wing under two conditions: the continuum flow and the slip flow. The following is shown. Within the range of Kn considered in this study (0.01 ≤ Kn ≤ 0.1), the rarefaction has a very small effect on the aerodynamic force of the bristled wing: it decreases the aerodynamic force by less than 0.5% compared with that of the continuum flow. However, the rarefaction has a significant effect on the contributions of the viscous tangential and normal stress terms to the aerodynamic force: in the continuum flow, the force contribution of the viscous tangential stress is 50.7% and that of the viscous normal stress is zero, whereas in the slip flow, e.g., at Kn = 0.08, the contribution of the viscous tangential stress is only 37.7% and that of the viscous normal stress is 12.9% instead of zero; this is because the rarefaction-induced slip velocity in the slip flow changes the normal derivative of the velocity on the bristle surface compared with that of the continuum flow. Since the rarefaction has only a slight effect on the aerodynamic force, the results on the aerodynamic force of the bristled wing obtained based on continuum flows in previous studies are very good approximations to the correct results.

Construction of a peridynamic model for viscous flow

We derive the Eulerian formulation for a peridynamic (PD) model of Newtonian viscous flow starting from fundamental principles: conservation of mass and momentum. This formulation is nonlocal, different from viscous flow models that utilize numerical methods like, e.g., the so-called “peridynamic differential operator” to approximate solutions of the classical Navier-Stokes equations. We show that the classical continuity equation is a limiting case of the PD one, assuming certain smoothness conditions. The PD model for viscous flow is calibrated by enforcing linear consistency for the viscous stress term with the classical Navier-Stokes equations. Couette and Poiseuille flows, and incompressible fluid flow past a regular lattice of cylinders are used to verify the new formulation, at low Reynolds numbers. The constructive approach in deriving the model allows for a seamless coupling with peridynamic models for corrosion or fracture for simulating complex fluid-structure interaction problems in which solid degradation takes place, such as in erosion-corrosion, hydraulic fracture, etc. Moreover, the new formulation sheds light on the relationships between local and nonlocal models.

Numerical simulation of non-Newtonian Carreau fluid in a lid-driven cavity

In this work, the 2D lid-driven cavity flow of non-Newtonian Carreau fluids has been studied by finite difference method on a staggered grid. A finite-difference algorithm on staggered grid based on projection method is adopted to solve the lid-driven cavity flow, which includes a second-order central difference scheme for the non-Newtonian viscous stress term. This study has been conducted for the certain pertinent parameters of Reynolds number (Re=100-1000), power-law index (n=0.6-1.4). The results show that as the Reynolds number increases, the influence of the power-law index on the flow increases. As the power-law index decreases, the flow field becomes more complicated.

Objective barriers to the transport of dynamically active vector fields

Abstract

Versatile mixed methods for the incompressible Navier–Stokes equations

In the spirit of the “Principle of Equipresence” introduced by Truesdell & Toupin, The Classical Field Theories (1960), we use the full version of the viscous stress tensor ν∇u+∇uT−23∇⋅uI which was originally derived for compressible flows, instead of the classical incompressible stress tensor ν∇u. (Note that, here ν is the dynamic viscosity coefficient, and u is the velocity field.) In our approach, the divergence-free constraint for the viscous stress term is not enforced ahead of discretization. Instead, our formulation allows the scheme itself to “choose” a consistent way to interpret the divergence-free constraint: i.e., the divergence-free constraint is interpreted (or enforced) in a consistent fashion in both the mass conservation equation and the stress tensor term (in the momentum equation). Furthermore, our approach preserves the original symmetrical properties of the stress tensor, e.g. its rotational invariance, and it remains physically correct in the context of compressible flows. As a result, our approach facilitates versatility and code reuse. In this paper, we introduce our approach and establish some important mathematical properties for the resulting class of finite element schemes. More precisely, for general mixed methods, which are not necessarily pointwise divergence-free, we establish the existence of a new norm induced by the full, viscous bilinear form. Thereafter, we prove the coercivity of the viscous bilinear form and the semi-coercivity of a convective trilinear form. In addition, we demonstrate L2-stability of the discrete velocity fields for the general class of methods and (by deduction) the H(div)-conforming methods. Finally, we run some numerical experiments to illustrate the behavior of the versatile mixed methods, and we make careful comparisons with a conventional H(div)-conforming scheme.

Evaluation of transient effects in the pressure-time method

The pressure-time is a method for measuring the flow rate in closed conduits and is typically used in hydropower applications. The scope of the present paper is to examine the physics of flow during a pressure-time measurement using experimental data and two-dimensional numerical simulations. The Unsteady Reynolds-averaged Navier–Stokes (URANS) equations coupled with the low-Re k-ω SST turbulence model are employed for the simulations. The contributions of inertia, pressure gradient, viscous and turbulent shear stress terms are investigated in the flow during a pressure-time measurement. It is shown that away from the wall and in the initial moments of time, the turbulent shear stress is comparable with the pressure gradient. With increasing time, the contribution of the inertia term becomes progressively significant and comparable with the pressure gradient and turbulent shear stress terms. Close to the wall, both viscous and turbulent shear stresses are the dominant terms which are diminished by increasing the time.

Computation of two- and three-dimensional water hammer flows

ABSTRACTThis paper investigates water hammer flows using two- and three-dimensional (2D and 3D) numerical simulations. The unsteady Reynolds-averaged Navier–Stokes (URANS) equations in conjunction with the k–ω SST turbulence model are employed for the computations. The valve modelling approach is used for 3D simulations, with superior agreement with the experiments. Similar predictions are obtained by 2D simulations and the flow rate reduction curve obtained from the 3D computations. The asymmetric flow patterns induced by the valve are confined within approximately one pipe diameter upstream of the valve. The contributions of inertia and pressure gradient terms are dominant at the instance of pressure wave passage, leading to abrupt changes in the fluid flow parameters. However, the effects of inertia and viscous shear stress terms are significant after the pressure wave passage, resulting in the flow tendency to approach a new steady condition. The viscous and turbulent dissipations are dominant close to and away from the wall, respectively.

Temporal discretization of viscous stress terms of incompressible Navier–Stokes equations with surface tension effect

In this study, four temporal discretization methods for viscous stress terms were examined through the finite element method for incompressible Navier–Stokes equations with surface tension effect. The temporal stability and numerical accuracy of the four methods were evaluated in static and rising bubble benchmark problems with severe temporal restrictions caused by viscous stress terms. First, conservative and non-conservative discretization methods were compared in terms of stability and accuracy. The stability and accuracy of the numerical solutions were further investigated through three temporal discretization methods (i.e., fully implicit, semi-implicit, and fully explicit) for viscous stress terms. Nonconservative discretization may yield an incorrect solution although it provides a compact element matrix that reduces CPU time for matrix–vector multiplication. Among the three temporal discretization methods for the conservative treatment of viscous stress terms, the fully implicit method is recommended for cases with strong viscous stress. This method requires an element matrix that is larger than those of the other methods. However, the time-step limit caused by the explicit or semiimplicit treatment of viscous stress terms can be avoided.

Direct numerical simulation of spatially developing turbulent boundary layers with opposition control

Opposition control of spatially developing turbulent boundary layers for skin friction drag reduction is studied by direct numerical simulations. The boundary layer extends 800θ0 in the streamwise (x) direction, with θ0 denoting the momentum thickness at the flow inlet. The Reynolds number, based on the external flow velocity and the momentum thickness, ranges from 300 to 860. Opposition control applied in different streamwise ranges, i.e. and as well as the uncontrolled case, are simulated. Statistical results and instantaneous flow fields are presented, with special attention paid to the spatial evolution properties of the boundary layer flow with control and the underlying mechanism. It is observed that a long spatial transient region after the control start and a long recovery region after the control end are present in the streamwise direction. A maximum drag reduction rate of about 22% is obtained as the transient region is passed, and an overshoot in the local skin friction coefficient (Cf) occurs in the recovery region. A new identity is derived for dynamical decomposition of Cf. Reduction of Cf by opposition control and overshoot of Cf in the recovery region are explained by quantifying the contributions from the viscous shear stress term, the Reynolds shear stress term, the mean convection term and other terms.

Stability of an annular power-law liquid sheet

Because of the mathematical difficulties dealing with the nonlinear viscous stress term in momentum equation for power-law liquid, the study of stability analysis of a power-law liquid sheet has been lacking. In the present study, a temporal stability analysis has been carried out for an annular power-law liquid sheet exposed to both inner and outer-gas streams, by integrating the governing equations for the power-law liquid sheet. The dimensionless dispersion equation that governs the instability of liquid sheet is obtained by considering the velocity profile of liquid sheet. It is found that the instability of liquid sheet can be enhanced by independently increasing either the outer gas stream velocity, or the inner gas stream velocity. The liquid sheet is more unstable when both inner and outer gas streams are applied. To promote the instability of annular liquid sheet, a gas stream applied to the outer interface is more effective than when applied to the inner surface. The effects of rheological parameters on the instability of the liquid sheet are actually determined by the relative velocity across the gas–liquid interfaces. The surface tension, liquid sheet thickness, and outer surface radius of annular liquid sheet have been tested for their influence on the instability of annular power-law liquid sheet.

Energy dissipation in unsteady turbulent pipe flows caused by water hammer

Energy dissipation and turbulent kinetic energy production and its dissipation in unsteady turbulent pipe flows due to water hammer phenomena are numerically studied. For this purpose, the two-dimensional governing equations of water hammer are solved using the method of characteristics. A k–ω turbulence model which is accurate for two-dimensional boundary layers under adverse and favorable pressure gradients is applied. The numerical results are in good agreement with the experimental data. Through an order of magnitude analysis, two dimensionless parameters have been identified which can be used for the evaluation of viscous and turbulent shear stress terms. The influence of these non-dimensional parameters on pressure oscillations, wall-shear-stress, dissipation rate as well as profiles of velocity, turbulent production and dissipation are investigated. The non-dimensional parameter P, which represents time scale ratio of turbulence diffusion in the radial direction to the pressure wave speed, is used to study the structure and strength of turbulence. It is found that for the case of P≈1, for which the values of the non-dimensional groups are larger, the peaks of turbulence energy production and dissipation move rapidly away from the wall and turbulence structure is significantly changed. For the case of P≫1, for which the values of non-dimensional parameters are smaller, these variations are found to be small.

On the Effects of Viscosity on the Shock Waves for a Hydrodynamical Case—Part I: Basic Mechanism

The interaction of shock waves with viscosity is one of the central problems in the supersonic regime of compressible fluid flow. In this work, numerical solutions of unmagnetised fluid equations, with the viscous stress tensor, are investigated for a one-dimensional shock wave. In the algorithm developed the viscous stress terms are expressed in terms of the relevant Reynolds number. The algorithm concentrated on the compression rate, the entropy change, pressures, and Mach number ratios across the shock wave. The behaviour of solutions is obtained for the Reynolds and Mach numbers defining the medium and shock wave in the supersonic limits.

Analysis of secondary current effect in the modeling of shallow flow in open channels

The 3D information of a secondary flow evolved in curved or confluent channels, which is usually lost in conventional shallow water models due to the depth-averaging process, can be partially retrieved by incorporating the secondary current velocity profile in the vertical direction. In the present study, the depth-averaged shallow water equations with dispersion stresses were solved by the SU/PG scheme among the various numerical methods of the FEM. In the proposed model, to quantify the strength of the secondary flow, the comprehensive factors such as the deformation of the vertical profile of the streamwise velocity, the roughness coefficient, and the curvature ratio were embedded into the dispersion stress terms. Two sets of experimental data, one with a sharply curved channel by Rozovskii (1961) [40], and the other with a confluent channel by Shumate (1998) [51] were used to validate the proposed model. The computed values of the water surface profile and the depth-averaged velocity across the channel showed good agreement with experimental data, which indicated that secondary velocity profiles were preserved properly. The proposed model was applied to a natural stream with moderate curvature to test the field applicability. The simulation results obtained by the proposed model with dispersion terms matched quite well with the ADCP data, whereas the model without dispersion terms produced excessive velocities at both banks, and the commercial RMA-2 model yielded uniform span-wise velocity distributions at all sections. The analysis of the momentum balance and relative dominance of the secondary current demonstrated that the convection and pressure gradient terms played major roles in achieving equilibrium in momentum equations, and the bottom friction ranked next, followed by the remaining dispersion stresses and viscous term of a similar order. It was also found that the pressure gradient term was the primary factor that triggered velocity redistribution. As the Fr increased, the convective acceleration that formed along the channel curvature was activated as a secondary factor in velocity redistribution, whereas the viscous stress term lost its influence. The bottom friction term had minor exertion on the redistribution mechanism.

Robustness and accuracy of SPH formulations for viscous flow

AbstractNumerous methods are available for the modelling of viscous stress terms in smoothed particle hydrodynamics (SPH). In this work, the existing methods are investigated systematically and evaluated for a range of Reynolds numbers using Poiseuille channel and lid‐driven cavity test cases. The best results are obtained using two methods based on combinations of finite difference and SPH approximations, due to Morris et al. and Cleary. Gradients of high‐valued functions are shown to be inaccurately estimated with standard SPH. A method that reduces the value of functions (in particular, pressure) before calculating the gradients reduces this inaccuracy and is shown to improve performance. A mode of instability in Poiseuille channel flows, also reported in other works, is examined and a qualitative explanation is proposed. The choice of boundary implementation is shown to have a significant effect on transient velocity profiles in start‐up of the flow. The use of at least linear extrapolation for in‐wall velocities is shown to be preferable to mirroring of velocities. Consistency corrections to the kernel are also found to result in significant accuracy and stability improvements with most methods, though not in all cases. Copyright © 2008 John Wiley & Sons, Ltd.

Boundary-layer analysis for effects of viscosity of the irrotational flow on the flow induced by a rapidly rotating cylinder in a uniform stream

We study the streaming flow past a rapidly rotating circular cylinder. The starting point is the full continuity and momentum equations without any approximations. We assume that the solution is a boundary-layer flow near the cylinder surface with the potential flow outside the boundary layer. The order of magnitude of the terms in the continuity and momentum equations can be estimated inside the boundary layer. When terms of the order of is the circulatory velocity at the outer edge of the boundary layer. The pressure from this boundary-layer solution has two parts, an inertial part and a viscous part. The inertial part comes from the inertia terms in the momentum equations and is in agreement with the irrotational pressure; the viscous part comes from the viscous stress terms in the momentum equations and may be viewed as a viscous pressure correction, which contributes to both drag and lift. Our boundary-layer solution is in reasonable to excellent agreement with the numerical simulation in the companion paper by Padrino & Joseph (2006).

Consistent large-eddy simulation of a temporal mixing layer laden with evaporating drops. Part 1. Direct numerical simulation, formulation and a priori analysis

Large-eddy simulation (LES) models are presented and evaluated on a database obtained from direct numerical simulation (DNS) of a three-dimensional temporal mixing layer with evaporating drops. The gas-phase equations are written in an Eulerian frame for two perfect gas species (carrier gas and vapour emanating from the drops), while the liquid-phase equations are written in a Lagrangian frame. The effect of drop evaporation on the gas phase is considered through mass, momentum and energy source terms. The DNS database consists of transitional states attained by layers with different initial Reynolds numbers and initial liquid-phase mass loadings. Budgets of the LES equations at the transitional states show that, for the mass loadings considered, the filtered source terms (FSTs) are smaller than the resolved inviscid terms and some subgrid scale (SGS) terms, but larger than the resolved viscous stress, heat flux and mass flux terms. The irreversible entropy production (i.e. the dissipation) expression for a two-phase flow with phase change is derived, showing that the dissipation contains contributions due to viscous stresses, heat and species-mass fluxes, and source terms. For both the DNS and filtered flow fields at transition, the two leading contributions are found to be the dissipation due to the energy source term and that due to the chemical potential of the mass source. Therefore, the modelling effort is focused on both the SGS fluxes and the FSTs in the LES equations. The FST models considered are applicable to LES in which the grid is coarser than the DNS grid and, consistently, ‘computational’ drops represent the DNS physical drops. Because the unfiltered flow field is required for the computation of the source terms, but would not be available in LES, it was approximated using the filtered flow field or the filtered flow field augmented by corrections based on the SGS variances. All of the FST models were found to overestimate DNS-field FSTs, with the relative error of modelling the unfiltered flow field compared to the error of using computational drops showing a complex dependence on filter width and number of computational drops. For modelling the SGS fluxes and (where possible) SGS variances, constant-coefficient Smagorinsky, gradient and scale-similarity models were assessed on the DNS database, and calibrated coefficients were statistically equivalent when computed on single-phase or two-phase flows. The gradient and scale-similarity models showed excellent correlation with the SGS quantities. An a posteriori study is proposed to evaluate the impact of the studied models on the flow-field development, so as to definitively assess their suitability for LES with evaporating drops.

The effect of viscous heating on the stability of Taylor–Couette flow

The influence of viscous heating on the stability of Taylor–Couette flow is investigated theoretically. Based on a linear stability analysis it is shown that viscous heating leads to significant destabilization of the Taylor–Couette flow. Specifically, it is shown that in the presence of viscous dissipation the most dangerous disturbances are axisymmetric and that the temporal characteristic of the secondary flow is very sensitive to the thermal boundary conditions. If the temperature difference between the two cylinders is small, the secondary flow is stationary as in the case of isothermal Taylor–Couette flow. However, when the temperature difference between the two cylinders is large, time-dependent secondary states are predicted. These linear stability predictions are in agreement with the experimental observations of White & Muller (2000) in terms of onset conditions as well as the spatiotemporal characteristics of the secondary flow. Nonlinear stability analysis has revealed that over a broad range of operating conditions, the bifurcation to the time-dependent secondary state is subcritical, while stationary states result as a consequence of supercritical bifurcation. Moreover, the supercritically bifurcated stationary state undergoes a secondary bifurcation to a time-dependent flow. Overall, the structure of the time-dependent state predicted by the analysis compares very well with the experimental observations of White & Muller (2000) that correspond to slowly moving vortices parallel to the cylinder axis. The significant destabilization observed in the presence of viscous heating arises as the result of the coupling of the perturbation velocity and the base-state temperature gradient that gives rise to fluctuations in the radial temperature distribution. Due to the thermal sensitivity of the fluid these fluctuations greatly modify the fluid viscosity and reduce the dissipation of disturbances provided by the viscous stress terms in the momentum equation.

Hysteresis and Stick-Slip Motion of Phase Boundaries in Dynamic Models of Phase Transitions

We investigate hysteretic behavior in two dynamic models for solid-solid phase transitions. An elastic bar with a nonconvex double-well elastic energy density is subjected to time-dependent displacement boundary conditions. Both models include inertia and a viscous stress term that provides energy dissipation. The first model involves a strain-gradient term that models interfacial energy. In the second model this term is omitted. Numerical simulations combined with analytical results predict hysteretic behavior in the overall end-load versus end-displacement diagram for both models. The hysteresis is largely due to metastability and nucleation; it persists even for very slow loading when viscous dissipation is quite small. In the model with interfacial energy, phase interfaces move smoothly. When this term is omitted, hysteresis is much more pronounced. In addition, phase boundaries move in an irregular, stick-slip fashion. The corresponding load-elongation curve exhibits serrations, in qualitative agreement with certain experimental observations in shape-memory alloys.

On the modelling of a PIB/PB Boger fluid in extensional flow

In this paper we study the transient elongational viscosity of a PIB/PB Boger fluid (fluid B) by performing both experiments and model calculations. The experimental results have been obtained by using a filament stretching device was described by van Nieuwkoop and Muller von Czernicki (J. Non-Newtonian Fluid Mech. 67 (1996) 105–123). We have been able to obtain large strains, up to a Hencky strain of 8. A plateau for the Trouton ratio of close to 10 4 is found while approaching a strain of 8. The plateau value appears to be independent of the strain rate history. The predictions using various multi-mode constitutive models (Giesekus, FENE-P, Hinch, FENE) show that the FENE model is the only model which predicts values for the Trouton ratio that are very close to the experiments for the whole range from strain 1 up to 8. The predictions of the models are computed with the actual strain rate ε ̇ t as determined during the experiments in the middle of the filament. During relaxation, both the FENE-P and the FENE model perform well. In order to be able to make predictions without carrying out the relatively expensive FENE calculations, we have developed a new closed form constitutive equation. The model is based on the dumbbell theory in which the connector force used leads to a viscous stress term. The predictions of the model in transient extensional flows are very good and comparable to the FENE model.