Variable Viscosity Flows Research Articles

High-order conservative formulation of viscous terms for variable viscosity flows

The work presents a general strategy to design high-order conservative co-located finite-difference approximations of viscous/diffusion terms for flows featuring extreme variations of diffusive properties. The proposed scheme becomes equivalent to central finite-difference derivatives with corresponding order in the case of uniform flow properties, while in variable viscosity/diffusion conditions it grants a strong preservation and a proper telescoping of viscous/diffusion terms. Presented tests show that standard co-located discretisation of the viscous terms is not able to describe the flow when the viscosity field experiences substantial variations, while the proposed method always reproduces the correct behaviour. Thus, the process is recommended for such flows whose viscosity field highly varies, in both laminar and turbulent conditions, relying on a more robust approximation of diffuse terms in any situation. Hence, the proposed discretisation should be used in all these cases and, for example, in large eddy simulations of turbulent wall flows where the eddy viscosity abruptly changes in the near-wall region.

Mixing behavior in a confined jet with disparate viscosity and implications for complex reactions

Turbulent shear-driven mixing in a coaxial and co-flowing configuration is studied using experiments and computations to understand and model the effect of viscosity gradients in the flow field. Two liquids with a large disparity in dynamic viscosity are mixed, with a low viscosity, high momentum jet directed into a high viscosity, low momentum co-flow in a pipe. Simultaneous experimental measurements of the velocity and concentration fields are made using high-resolution PIV and PLIF to obtain their turbulent cross-correlation statistics for viscosity ratios of 1 and 40. LES simulations are also performed using dynamic mixing sub-grid model to investigate the three dimensional mixing structure of the flow for the two cases. The overall structure of turbulent mixing in the coaxial confined jet configuration is studied to identify the mixing regions of the flow and the effect of viscosity gradients on the dynamics of the same. Besides the effect of Reynolds number between the two cases that manifests as reduced mixing, it was noted that the transport of turbulent kinetic energy and scalar concentration variance shows significant asymmetries that arise from the viscosity gradients in the field. The scalar mixing structure between the two cases is studied in detail with relevance to complex mixing-limited reactions frequently encountered in such environments. It was found that turbulence production and associated scalar mixing is highly skewed in variable viscosity flows, where the low viscosity regions show enhanced turbulence activity. The implications of such turbulence skewness on the chemistry of reaction systems involving variable viscosity environments are discussed in further detail.

Self-similarity criteria in anisotropic flows with viscosity stratification

Variable-viscosity flows exhibit a faster trend towards a fully developed turbulent state since fluctuations are produced at a larger amount. A legitimate expectation is that self-similarity to be tenable earlier than in classical, single-viscosity flows. The question which begs to be answered is: which are the self-similarity criteria for variable-viscosity, density-matched, flows? The similarity assumption, i.e., all scales evolve in a similar fashion in space/time, is applied to the transport equation for one- and two-point statistics of anisotropic, variable-viscosity flows. It is shown that the similarity assumption is valid for regions of the flow where viscosity (mean values and the fluctuations root-mean-square) is uniform. In regions where viscosity gradients are important, such as the sheared region and jet boundaries, similarity is not tenable. Our claims are applicable to any decaying flow, isotropic or anisotropic. Support is provided by experimental data obtained in the near field region of a jet issuing into a more viscous environment. The viscosity ratio is 3.5.

On the similarity of variable viscosity flows

Turbulent mixing is ubiquitous in both nature and industrial applications. Most of them concern different fluids, therefore with variable physical properties (density and/or viscosity). The focus here is on variable viscosity flows and mixing, involving density-matched fluids. The issue is whether or not these flows may be self-similar, or self-preserving. The importance of this question stands on the predictability of these flows; self-similar dynamical systems are easier tractable from an analytical viewpoint. More specifically, self-similar analysis is applied to the scale-by-scale energy transport equations, which represent the transport of energy at each scale and each point of the flow. Scale-by-scale energy budget equations are developed for inhomogeneous and anisotropic flows, in which the viscosity varies as a result of heterogeneous mixture or temperature variations. Additional terms are highlighted, accounting for the viscosity gradients, or fluctuations. These terms are present at both small and large scales, thus rectifying the common belief that viscosity is a small-scale quantity. Scale-by-scale energy budget equations are then adapted for the particular case of a round jet evolving in a more viscous host fluid. It is further shown that the condition of self-preservation is not necessarily satisfied in variable-viscosity jets. Indeed, the jet momentum conservation, as well as the constancy of the Reynolds number in the central region of the jet, cannot be satisfied simultaneously. This points to the necessity of considering less stringent conditions (with respect to classical, single-fluid jets) when analytically tackling these flows and reinforces the idea that viscosity variations must be accounted for when modelling these flows.

Turbulent thermal boundary layers with temperature-dependent viscosity

Direct numerical simulations (DNS) of turbulent boundary layers (TBLs) over isothermally heated walls were performed, and the influence of the wall-heating on the thermal boundary layers was investigated. The DNS adopt an empirical relation for the temperature-dependent viscosity of water. The Prandtl number therefore changes with temperature, while the Péclet number is constant. Two wall temperatures (Tw=70°C and 99°C) were considered relative to T∞=30°C, and a reference simulation of TBL with constant viscosity was also performed for comparison. In the variable viscosity flow, the mean and variance of the scalar, when normalized by the friction temperature deficit, decrease relative to the constant viscosity flow. A relation for the mean scalar which takes into account the variable viscosity is proposed. Appropriate scalings for the scalar fluctuations and the scalar flux are also introduced, and are shown to be applicable for both variable and constant viscosity flows. Due to the modification of the near-wall turbulence, the Stanton number and the Reynolds analogy factor are augmented by 10% and 44%, respectively, in the variable viscosity flow. An identity for the Stanton number is derived and shows that the mean wall-normal velocity and wall-normal scalar flux cause the increase in the heat transfer coefficient. Finally, the augmented near-wall velocity fluctuations lead to an increase of the wall-normal scalar flux, which contributes favorably to the enhanced heat transfer at the wall.

Temperature-dependent shear flow and the absence of thermal runaway in valley glaciers

We propose a two-dimensional model of a valley glacier in order to reconsider the question of whether thermal runaway could be a viable mechanism for the onset of creep instability in surging glaciers. We do this by providing an approximate solution for the temperature field based on the idea that shear is concentrated at the glacier bed. With this assumption, we show that a closed-form evolution equation for the glacier profile exists. While this is well known for isoviscous flows, it has not been previously derived for variable viscosity flows. During the process of deriving this equation, we show that thermal runaway does not occur. We provide numerical solutions of the model, and are led to infer that enhanced basal heating owing to refreezing of surface meltwater is an essential constituent in raising the bed temperature to the melting point.

Time-resolved velocity and concentration measurements in variable-viscosity turbulent jet flow

This paper addresses the ability to reliably measure the fluctuating velocity field in variable-viscosity flows (herein, a propane–air mixture), using hot-wire anemometry. Because the latter is sensitive to both velocity and concentration fluctuations, the instantaneous concentration field also needs to be inferred experimentally. To overcome this difficulty, we show that the hot-wire response becomes insensitive to the concentration of the field, when a small amount of neon is added to the air. In this way, velocity measurements can be made independently of the concentration field. Although not necessary to velocity measurements, Rayleigh light-scattering technique is also used to infer the local (fluctuating) concentration, and, therefore, the viscosity of the fluid. Velocity and concentration measurements are performed in a turbulent propane jet discharging into an air–neon co-flow, for which the density and viscosity ratios are 1.52 and 1/5.5, respectively. The Reynolds number (based on injection diameter and velocity) is 15400. These measurements are first validated: the axial decay of the mean velocity and concentration, as well as the lateral mean and RMS profiles of velocity and concentration, is in full agreement with the existing literature. The variable-viscosity flow along the axis of the round jet is then characterized and compared with a turbulent air jet discharging into still air, for which the Reynolds number (based on injection diameter and velocity) is 5400. Both flows have the same initial jet momentum. As mixing with the viscous co-flow is enhanced with increasing downstream position, the viscosity of the fluid increases rapidly for the case of the propane jet. In comparison with the air jet, the propane jet exhibits: (1) a lower local Reynolds number based on the Taylor microscale (by a factor of four); (2) a reduced range of scales present in the flow; (3) the isotropic form of the mean energy dissipation rate is first more enhanced and then drastically diminishes and (4) a progressively increasing local Schmidt number (from 1.36 to 7.5) for increasing downstream positions. Therefore, the scalar spectra exhibit an increasingly prominent Batchelor regime with a ~ k −1 scaling law. The experimental technique developed herein provides a reliable method for the study of variable-viscosity flows.

Error assessment of lattice Boltzmann equation method for variable viscosity flows

AbstractIn lattice Boltzmann simulations, variable viscosity can complicate the truncation error analysis and create additional interaction between the truncation error and the boundary condition error. In order to address this issue, two boundary conditions for the lattice Boltzmann equation (LBE) simulations are used, including an exact, but narrowly applicable scheme previously proposed by Noble et al. (Phys. Fluids 1995; 7(1):203–209) and the popular bounce‐back‐on‐link scheme. Using a 2‐D laminar channel flow with a specified variable viscosity as a test case, it is shown that the boundary treatment error does not have a significant interaction with the truncation error associated with variable viscosity. The truncation error behaviour of the LBE for flows with variable viscosity is further investigated through a comparison between the LBE solution and the Navier–Stokes solution, showing that in the presence of strong variable viscosity the truncation error behaviour of the LBE solution is consistent with that of the Navier–Stokes solution, indicating that the LBE model closely matches the Navier–Stokes model for fluid flows with large viscosity variation. Copyright © 2006 John Wiley & Sons, Ltd.

DIRECT SPECTRAL METHODS FOR THE LOW MACH NUMBER EQUATIONS

The low Mach number approximation of the Navier—Stokes equations is of similar nature to the equations for incompressible flow. A major difference, however, is the appearance of a space‐ and time‐varying density that introduces a supplementary non‐linearity. In order to solve these equations with spectral space discretization, an iterative solution method has been constructed and successfully applied in former work to two‐dimensional natural convection and isobaric combustion with one direction of periodicity. For the extension to other geometries efficiency is an important point, and it is therefore desirable to devise a direct method which would have, in the best case, the same stability properties as the iterative method. The present paper discusses in a systematic way different approaches to this aim. It turns out that direct methods avoiding the diffusive time step limit are possible, indeed. Although we focus for discussion and numerical investigation on natural convection flows, the results carry over for other problems such as variable viscosity flows, isobaric combustion, or non‐homogeneous flows.

Wave-like transitions and stability in variable-viscosity flows

This work deals with the simplest one-dimensional formulation of variable-viscosity isothermal flow in a tube (tubular reactor). The case of rising viscosity of reacting (polymerizing) fluid is considered, where there are three solutions corresponding to one value of external setting pressure drop. The stability and transitions when the setting pressure drop is varied are investigated.

On the thermal runaway of variable viscosity flows between concentric cylinders

In this paper we use asymptotic methods to analyse the boundary layer flow of a Newtonian fluid between concentric cylinders. The temperature distribution up to the time of thermal runaway is obtained. We give conditions for occurrence of thermal runaway.

Variable-viscosity flows in channels with high heat generation

This paper presents a similarity solution for plane channel flow of a very viscous fluid, whose viscosity is exponentially dependent upon temperature, when heat generation is very large. A dimensionless formulation of the problem involves two length scales (the depthhand lengthl, respectively, of the channel), one velocity scale (the mean velocityVof the fluid along the channel), the thermal conductivityk, thermal diffusivitykand viscosityVof the fluid, and the temperature coefficientbof the viscosity. From these, two important dimensionless groups arise, the Graetz number (Gz = Vh2/kl) and the Nahme–Griffith number (G= μV2b/k). In the case of steady flow withG−1[Lt ]Gz−1[Lt ] 1 a thin thermal boundary layer of thickness proportional toGz−½arises at each wall with an even thinner shear layer, detached from the wall and embedded in the thermal boundary layer, of thickness proportional toGz−½(lnG)−1, coinciding with the region of maximum temperature (lnG)/b. The similarity variable is (Pe½y/x½) wherePeis the Péclet number (Vh/k) andyandxare measured away from and along (either) boundary wall. The analogous unsteady uniform flow solution is also given.

Variable-viscosity flows in heated and cooled channels

An asymptotic description is given of Newtonian fluid flow in a channel which is suddenly heated or cooled. The viscosity is assumed to be purely a function of temperature. The asymptotic approximation is that the downstream viscosity at the channel wall differs by an order of magnitude from that in the upstream flow. Although we make the drastic assumption that viscous dissipation is negligible, we can analyse flows where the viscosity depends either algebraically or exponentially on the temperature.

Numerical experiments on the structure of mantle plumes

Numerical experiments on steady state thermal convection in a cylindrical geometry have been carried out to identify the conditions necessary for plumes to occur in the mantle. Internally and base-heated flows with both constant and variable diffusion creep viscosity have been considered. These experiments show that plumes occur only for base-heated variable viscosity flows. Plumes are examined as a possible mode of convection in the upper mantle above 700 km, since it is in this region of the mantle that base heating should be most dominant. The consequences of an upper mantle plume under Hawaii are discussed in terms of available geological and geophysical observations, and good agreement is found. Plumes also explain the shape of the geotherm under South Africa derived by pyroxene geothermometry.