The role of bulk viscosity on the decay of compressible, homogeneous, isotropic turbulence

2017 ◽  
Vol 833 ◽  
pp. 717-744 ◽  
Author(s):  
Shaowu Pan ◽  
Eric Johnsen
Keyword(s):  
Energy Transfer ◽  
Turbulent Flows ◽  
Bulk Viscosity ◽  
Shear Viscosity ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Compressible Turbulence ◽  
Homogeneous Isotropic Turbulence ◽  
Viscous Effects ◽  
Bulk Viscous

While Stokes’ hypothesis of neglecting bulk viscous effects is exact for monatomic gases and unlikely to strongly affect the dynamics of fluids whose bulk-to-shear viscosity ratio is small and/or of weakly compressible turbulence, it is unclear to what extent this assumption holds for compressible, turbulent flows of gases whose bulk viscosity is orders of magnitude larger than their shear viscosities (e.g. $\text{CO}_{2}$). Our objective is to understand the mechanisms by which bulk viscosity and the associated phenomena affect moderately compressible turbulence, in particular energy transfer and dissipation. Using direct numerical simulations of the compressible Navier–Stokes equations, we study the decay of compressible, homogeneous, isotropic turbulence for ratios of bulk-to-shear viscosity ranging from 0 to 1000. Our simulations demonstrate that bulk viscosity increases the decay rate of turbulent kinetic energy; whereas enstrophy exhibits little sensitivity to bulk viscosity, dilatation is reduced by over two orders of magnitude within the first two eddy-turnover times. Via a Helmholtz decomposition of the flow, we determine that the primary action of bulk viscosity is to damp the dilatational velocity fluctuations and reduce dilatational–solenoidal exchanges, as well as pressure–dilatation coupling. In short, bulk viscosity renders compressible turbulence incompressible by reducing energy transfer between translational and internal degrees of freedom. Our results indicate that for gases whose bulk viscosity is of the order of their shear viscosity (e.g. hydrogen) the turbulence is not significantly affected by bulk viscous dissipation, in which case neglecting bulk viscosity is acceptable in practice. However, in problems involving compressible, turbulent flows of gases like $\text{CO}_{2}$ whose bulk viscosities are thousands of times greater than their shear viscosities, bulk viscosity cannot be ignored.

scholarly journals Higher-order dissipation in the theory of homogeneous isotropic turbulence

2016 ◽  
Vol 803 ◽  
pp. 250-274 ◽  
Author(s):  
Norbert Peters ◽  
Jonas Boschung ◽  
Michael Gauding ◽  
Jens Henrik Goebbert ◽  
Reginald J. Hill ◽  
...  
Keyword(s):  
Source Term ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Higher Order ◽  
Distribution Functions ◽  
Order Structure ◽  
Source Terms ◽  
Homogeneous Isotropic Turbulence ◽  
Higher Order Moments ◽  
Even Order

The two-point theory of homogeneous isotropic turbulence is extended to source terms appearing in the equations for higher-order structure functions. For this, transport equations for these source terms are derived. We focus on the trace of the resulting equations, which is of particular interest because it is invariant and therefore independent of the coordinate system. In the trace of the even-order source term equation, we discover the higher-order moments of the dissipation distribution, and the individual even-order source term equations contain the higher-order moments of the longitudinal, transverse and mixed dissipation distribution functions. This shows for the first time that dissipation fluctuations, on which most of the phenomenological intermittency models are based, are contained in the Navier–Stokes equations. Noticeably, we also find the volume-averaged dissipation $\unicode[STIX]{x1D700}_{r}$ used by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) in the resulting system of equations, because it is related to dissipation correlations.

The analysis and modelling of dilatational terms in compressible turbulence

1991 ◽  
Vol 227 ◽  
pp. 473-493 ◽  
Author(s):  
S. Sarkar ◽  
G. Erlebacher ◽  
M. Y. Hussaini ◽  
H. O. Kreiss
Keyword(s):  
Asymptotic Analysis ◽  
Numerical Simulations ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Mixing Layer ◽  
Direct Numerical Simulations ◽  
Homogeneous Turbulence ◽  
Compressible Turbulence ◽  
Quasi Equilibrium ◽  
Compressible Mixing Layer

It is shown that the dilatational terms that need to be modelled in compressible turbulence include not only the pressure-dilatation term but also another term - the compressible dissipation. The nature of the compressible velocity field, which generates these dilatational terms, is explored by asymptotic analysis of the compressible Navier-Stokes equations in the case of homogeneous turbulence. The lowest-order equations for the compressible field are solved and explicit expressions for some of the associated one-point moments are obtained. For low Mach numbers, the compressible mode has a fast timescale relative to the incompressible mode. Therefore, it is proposed that, in moderate Mach number homogeneous turbulence, the compressible component of the turbulence is in quasi-equilibrium with respect to the incompressible turbulence. A non-dimensional parameter which characterizes this equilibrium structure of the compressible mode is identified. Direct numerical simulations (DNS) of isotropic, compressible turbulence are performed, and their results are found to be in agreement with the theoretical analysis. A model for the compressible dissipation is proposed; the model is based on the asymptotic analysis and the direct numerical simulations. This model is calibrated with reference to the DNS results regarding the influence of compressibility on the decay rate of isotropic turbulence. An application of the proposed model to the compressible mixing layer has shown that the model is able to predict the dramatically reduced growth rate of the compressible mixing layer.

Effect of Schmidt number on small-scale passive scalar turbulence

2003 ◽  
Vol 56 (6) ◽  
pp. 615-632 ◽  
Author(s):  
RA Antonia ◽  
P Orlandi
Keyword(s):  
Turbulent Flows ◽  
Schmidt Number ◽  
Isotropic Turbulence ◽  
Passive Scalar ◽  
Scalar Dissipation ◽  
Small Scale ◽  
Homogeneous Isotropic Turbulence ◽  
Passive Scalars ◽  
First Case ◽  
Decaying Turbulence

Previous reviews of the behavior of passive scalars which are convected and mixed by turbulent flows have focused primarily on the case when the Prandtl number Pr, or more generally, the Schmidt number Sc is around 1. The present review considers the extra effects which arise when Sc differs from 1. It focuses mainly on information obtained from direct numerical simulations of homogeneous isotropic turbulence which either decays or is maintained in steady state. The first case is of interest since it has attracted significant theoretical attention and can be related to decaying turbulence downstream of a grid. Topics covered in the review include spectra and structure functions of the scalar, the topology and isotropy of the small-scale scalar field, as well as the correlation between the fluctuating rate of strain and the scalar dissipation rate. In each case, the emphasis is on the dependence with respect to Sc. There are as yet unexplained differences between results on forced and unforced simulations of homogeneous isotropic turbulence. There are 144 references cited in this review article.

Studying turbulence using direct numerical simulation: 1987 Center for Turbulence Research NASA Ames/Stanford Summer Programme

1988 ◽  
Vol 190 ◽  
pp. 375-392 ◽  
Author(s):  
J. C. R. Hunt
Keyword(s):  
Boundary Layers ◽  
Turbulent Flows ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Turbulent Boundary Layers ◽  
Finite Amplitude ◽  
Wall Layer ◽  
Navier Stokes ◽  
Scale Motion

This paper is an account of a summer programme for the study of the ideas and models of turbulent flows, using the results of direct numerical stimulations of the Navier-Stokes equations. These results had been obtained on the computers and stored as accessible databases at the Center for Turbulence Research (CTR) of NASA Ames Research Center and Stanford University. At this first summer programme, some 32 visiting researchers joined those at the CTR to test hypotheses and models in five aspects of turbulence research: turbulence decomposition, bifurcation and chaos; two-point closure (or k-space) modelling; structure of turbulent boundary layers; Reynolds-stress modelling; scalar transport and reacting flows.A number of new results emerged including: computation of space and space-time correlations in isotropic turbulence can be related to each other and modelled in terms of the advection of small scales by large-scale motion; the wall layer in turbulent boundary layers is dominated by shear layers which protrude into the outer layers, and have long lifetimes; some aspects of the ejection mechanism for these layers can be described in terms of the two-dimensional finite-amplitude Navier-Stokes solutions; a self-similar form of the two-point, cross-correlation data of the turbulence in boundary layers (when normalized by the r.m.s. value at the furthest point from the wall) shows how both the blocking of eddies by the wall and straining by the mean shear control the lengthscales; the intercomponent transfer (pressure-strain) is highly localized in space, usually in regions of concentrated vorticity; conditioned pressure gradients are linear in the conditioning of velocity and independent of vorticity in homogeneous shear flow; some features of coherent structures in the boundary layer are similar to experimental measurements of structures in mixing-layers, jets and wakes.The availability of comprehensive velocity and pressure data certainly helps the investigation of concepts and models. But a striking feature of the summer programme was the diversity of interpretation of the same computed velocity fields. There are few signs of any convergence in turbulence research! But with new computational facilities the divergent approaches can at least be related to each other.

Scale invariance in finite Reynolds number homogeneous isotropic turbulence

2019 ◽  
Vol 864 ◽  
pp. 244-272 ◽  
Author(s):  
L. Djenidi ◽  
R. A. Antonia ◽  
S. L. Tang
Keyword(s):  
Reynolds Number ◽  
Scale Invariance ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Longitudinal Velocity ◽  
Navier Stokes ◽  
Invariance Property ◽  
Homogeneous Isotropic Turbulence ◽  
Navier Stokes Equations ◽  
Flatness Factor

The problem of homogeneous isotropic turbulence (HIT) is revisited within the analytical framework of the Navier–Stokes equations, with a view to assessing rigorously the consequences of the scale invariance (an exact property of the Navier–Stokes equations) for any Reynolds number. The analytical development, which is independent of the 1941 (K41) and 1962 (K62) theories of Kolmogorov for HIT for infinitely large Reynolds number, is applied to the transport equations for the second- and third-order moments of the longitudinal velocity increment, $(\unicode[STIX]{x1D6FF}u)$. Once the normalised equations and the constraints required for complying with the scale-invariance property of the equations are presented, results derived from these equations and constraints are discussed and compared with measurements. It is found that the fluid viscosity, $\unicode[STIX]{x1D708}$, and the mean kinetic energy dissipation rate, $\overline{\unicode[STIX]{x1D716}}$ (the overbar denotes spatial and/or temporal averaging), are the only scaling parameters that make the equations scale-invariant. The analysis further leads to expressions for the distributions of the skewness and the flatness factor of $(\unicode[STIX]{x1D6FF}u)$ and shows that these distributions must exhibit plateaus (of different magnitudes) in the dissipative and inertial ranges, as the Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ increases indefinitely. Also, the skewness and flatness factor of the longitudinal velocity derivative become constant as $Re_{\unicode[STIX]{x1D706}}$ increases; this is supported by experimental data. Further, the analysis, backed up by experimental evidence, shows that, beyond the dissipative range, the behaviour of $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}$ with $n=2$, 3 and 4 cannot be represented by a power law of the form $r^{\unicode[STIX]{x1D701}_{n}}$ when the Reynolds number is finite. It is shown that only when $Re_{\unicode[STIX]{x1D706}}\rightarrow \infty$ can an $n$-thirds law (i.e. $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}\sim r^{\unicode[STIX]{x1D701}_{n}}$, with $\unicode[STIX]{x1D701}_{n}=n/3$) emerge, which is consistent with the onset of a scaling range.

A Markovian random coupling model for turbulence

1974 ◽  
Vol 65 (1) ◽  
pp. 145-152 ◽  
Author(s):  
U. Frisch ◽  
M. Lesieur ◽  
A. Brissaud
Keyword(s):  
Stochastic Model ◽  
Master Equation ◽  
Turbulent Flows ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Planck Equation ◽  
Navier Stokes ◽  
Coupling Coefficients ◽  
Navier Stokes Equations ◽  
Random Coupling

The Markovian random coupling (MRC) model is a modified form of the stochastic model of the Navier-Stokes equations introduced by Kraichnan (1958, 1961). Instead of constant random coupling coefficients, white-noise time dependence is assumed for the MRC model. Like the Kraichnan model, the MRC model preserves many structural properties of the original Navier-Stokes equations and should be useful for investigating qualitative features of turbulent flows, in particular in the limit of vanishing viscosity. The closure problem is solved exactly for the MRC model by a technique which, contrary to the original Kraichnan derivation, is not based on diagrammatic expansions. A closed equation is obtained for the functional probability distribution of the velocity field which is a special case of Edwards’ (1964) Fokker-Planck equation; this equation is an exact consequence of the stochastic model whereas Edwards’ equation constitutes only the first step in a formal expansion based directly on the Navier-Stokes equations. From the functional equation an exact master equation is derived for simultaneous second-order moments which happens to be essentially a Markovianized version of the single-time quasi-normal approximation characterized by a constant triad-interaction time.The explicit form of the MRC master equation is given for the Burgers equation and for two- and three-dimensional homogeneous isotropic turbulence.

Small-scale two-dimensional turbulence shaped by bulk viscosity

2019 ◽  
Vol 875 ◽  
pp. 974-1003
Author(s):  
Emile Touber
Keyword(s):  
Kinetic Energy ◽  
Bulk Viscosity ◽  
Shear Viscosity ◽  
Stokes Equations ◽  
Experimental Studies ◽  
Turbulence Kinetic Energy ◽  
Small Scale ◽  
Two Dimensional ◽  
Large Bulk ◽  
High Bulk

Bulk-to-shear viscosity ratios of three orders of magnitude are often reported in carbon dioxide but are always neglected when predicting aerothermal loads in external (Mars exploration) or internal (turbomachinery, heat exchanger) turbulent flows. The recent (and first) numerical investigations of that matter suggest that the solenoidal turbulence kinetic energy is in fact well predicted despite this seemingly arbitrary simplification. The present work argues that such a conclusion may reflect limitations from the choice of configuration rather than provide a definite statement on the robustness of kinetic-energy transfers to the use of Stokes’ hypothesis. Two distinct asymptotic regimes (Euler–Landau and Stokes–Newton) in the eigenmodes of the Navier–Stokes equations are identified. In the Euler–Landau regime, the one captured by earlier studies, acoustic and entropy waves are damped by transport coefficients and the dilatational kinetic energy is dissipated, even more rapidly for high bulk-viscosity fluids and/or forcing frequencies. If the kinetic energy is initially or constantly injected through solenoidal motions, effects on the turbulence kinetic energy remain minor. However, in the Stokes–Newton regime, diffused bulk compressions and advected isothermal compressions are found to prevail and promote small-scale enstrophy via vorticity–dilatation correlations. In the absence of bulk viscosity, the transition to the Stokes–Newton regime occurs within the dissipative scales and is not observed in practice. In contrast, at high bulk viscosities, the Stokes–Newton regime can be made to overlap with the inertial range and disrupt the enstrophy at small scales, which is then dissipated by friction. Thus, flows with substantial inertial ranges and large bulk-to-shear viscosity ratios should experience enhanced transfers to small-scale solenoidal kinetic energy, and therefore faster dissipation rates leading to modifications of the heat-transfer properties. Observing numerically such transfers is still prohibitively expensive, and the present simulations are restricted to two-dimensional turbulence. However, the theory laid here offers useful guidelines to design experimental studies to track the Stokes–Newton regime and associated modifications of the turbulence kinetic energy, which are expected to persist in three-dimensional turbulence.

Simulation of Interactions Between Microbubbles and Turbulent Flows

1994 ◽  
Vol 47 (6S) ◽  
pp. S70-S74 ◽  
Author(s):  
M. R. Maxey ◽  
E. J. Chang ◽  
L. -P. Wang
Keyword(s):  
Turbulent Flows ◽  
Isotropic Turbulence ◽  
Small Scale ◽  
Air Bubbles ◽  
Homogeneous Isotropic Turbulence ◽  
Mean Flow ◽  
Drag Forces ◽  
Spherical Inclusions ◽  
Average Rise ◽  
Surrounding Fluid

Microbubbles formed by small air bubbles in water are characterized as spherical inclusions that are essentially rigid due to the effects of surfactants, and respond to the action of drag forces and added-mass effects from the motion relative to the surrounding fluid. Direct numerical simulations of homogeneous, isotropic turbulence are used to study the effects of the small-scale, dissipation range turbulence on microbubble transport and in particular the average rise velocity of microbubbles. It is found that microbubbles rise significantly more slowly than in still fluid even in the absence of a mean flow, due to a strong interaction with the small-scale vorticity. The way in which microbubbles might modify the underlying turbulence by the variations in their local distribution is discussed for dilute, dispersed systems and some estimates for the enhanced viscous dissipation given.

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