Density fluctuation correlations in free turbulent binary mixing

1994 ◽  
Vol 279 ◽  
pp. 239-278 ◽  
Author(s):  
P. Chassaing ◽  
G. Harran ◽  
L. Joly
Keyword(s):  
Density Fluctuation ◽  
Fluid Motion ◽  
Variable Density ◽  
Second Order ◽  
Decay Rates ◽  
Entrainment Rate ◽  
Mean Velocity ◽  
Independent Variables ◽  
Binary Mixing ◽  
Reynolds Averaging

This paper is devoted to the analysis of the turbulent mass flux and, more generally, of the density fluctuation correlation (d.f.c.) effects in variable-density fluid motion. The situation is restricted to the free turbulent binary mixing of an inhomogeneous round jet discharging into a quiescent atmosphere. Based on conventional (Reynolds) averaging, a ternary regrouping of the correlations occurring in the statistical averaging of the open equations is first introduced. Then an exact algebraic relationship between the d.f.c. terms and the second-order moments is demonstrated. Some consequences of this result on the global behaviour of variable-density jets are analytically discussed. The effects of the d.f.c. terms are shown to give a qualitative explanation of the influence of the ratio of the densities of the inlet jet and ambient fluid on the centerline decay rates of mean velocity and mass fraction, the entrainment rate and the restructuring of the jet. Finally, the sensitivity of second-order modelling to the d.f.c. terms is illustrated and it is suggested that such terms should be considered as independent variables in the closing procedure.

Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density

2017 ◽  
Vol 6 (2) ◽  
pp. 121-145 ◽  
Author(s):  
Marcelo M. Cavalcanti ◽  
Valéria N. Domingos Cavalcanti ◽  
Irena Lasiecka ◽  
Claudete M. Webler
Keyword(s):  
Evolution Equations ◽  
Variable Density ◽  
Second Order ◽  
Decay Rates ◽  
Equation Modeling ◽  
Relaxation Kernel ◽  
Time Behavior ◽  
Content Type ◽  
Frictional Damping ◽  
Definite Operator

AbstractWe consider the long-time behavior of a nonlinear PDE with a memory term which can be recast in the abstract form$\frac{d}{dt}\rho(u_{t})+Au_{tt}+\gamma A^{\theta}u_{t}+Au-\int_{0}^{t}g(s)Au(t% -s)=0,$where A is a self-adjoint, positive definite operator acting on a Hilbert space H, ${\rho(s)}$ is a continuous, monotone increasing function, and the relaxation kernel ${g(s)}$ is a continuous, decreasing function in ${L_{1}(\mathbb{R}_{+})}$ with ${g(0)>0}$. Of particular interest is the case when ${A=-\Delta}$ with appropriate boundary conditions and ${\rho(s)=|s|^{\rho}s}$. This model arises in the context of solid mechanics accounting for variable density of the material. While finite energy solutions of the underlying PDE solutions exhibit exponential decay rates when strong damping is active (${\gamma>0,\theta=1}$), this uniform decay is no longer valid (by spectral analysis arguments) for dynamics subjected to frictional damping only, say, ${\theta=0}$ and ${g=0}$. In the absence of mechanical damping (${\gamma=0}$), the linearized version of the model reduces to a Volterra equation generated by bounded generators and, hence, it is exponentially stable for exponentially decaying kernels. The aim of the paper is to study intrinsic decays for the energy of the nonlinear model accounting for large classes of relaxation kernels described by the inequality ${g^{\prime}+H(g)\leq 0}$ with H convex and subject to the assumptions specified in (1.13) (a general framework introduced first in [1] in the context of linear second-order evolution equations with memory). In the context of frictional damping, such a framework was introduced earlier in [15], where it was shown that the decay rates of second-order evolution equations with frictional damping can be described by solutions of an ODE driven by a suitable convex function H which captures the behavior at the origin of the dissipation. The present paper extends this analysis to nonlinear equations with viscoelasticity. It is shown that the decay rates of the energy are intrinsically described by the solution of the dissipative ODE${S_{t}+c_{1}H(c_{2}S)=0}$with given intrinsic constants ${c_{1},c_{2}>0}$. The results obtained are sharp and they improve (by introducing a novel methodology) previous results in the literature (see [20, 19, 21, 6]) with respect to (i) the criticality of the nonlinear exponent ρ and (ii) the generality of the relaxation kernel.

Static pressure distribution in the free turbulent jet

1957 ◽  
Vol 3 (1) ◽  
pp. 1-16 ◽  
Author(s):  
David R. Miller ◽  
Edward W. Comings
Keyword(s):  
Static Pressure ◽  
Longitudinal Velocity ◽  
Fluid Motion ◽  
Mean Velocity ◽  
Free Jet ◽  
Static Pressure Distribution ◽  
Two Dimensional Flow ◽  
The Common ◽  
Jet Theory ◽  
Made In

Measurements of mean velocity, turbulent stress and static pressure were made in the mixing region of a jet of air issuing from a slot nozzle into still air. The velocity was low and the two-dimensional flow was effectively incompressible. The results are examined in terms of the unsimplified equations of fluid motion, and comparisons are drawn with the common assumptions and simplifications of free jet theory. Appreciable deviations from isobaric conditions exist and the deviations are closely related to the local turbulent stresses. Negative static pressures were encountered everywhere in the mixing field except in the potential wedge region immediately adjacent to the nozzle. Lateral profiles of mean longitudinal velocity conformed closely to an error curve at all stations further than 7 slot widths from the nozzle mouth. An asymptotic approach to complete self-preservation of the flow was observed.

scholarly journals Wake behind a three-dimensional dry transom stern. Part 1. Flow structure and large-scale air entrainment

2019 ◽  
Vol 875 ◽  
pp. 854-883 ◽  
Author(s):  
Kelli Hendrickson ◽  
Gabriel D. Weymouth ◽  
Xiangming Yu ◽  
Dick K.-P. Yue
Keyword(s):  
Froude Number ◽  
Flow Structure ◽  
Large Scale ◽  
Three Dimensional ◽  
Air Entrainment ◽  
Breaking Waves ◽  
Variable Density ◽  
Entrainment Rate ◽  
Two Phase ◽  
Density Spectrum

We present high-resolution implicit large eddy simulation (iLES) of the turbulent air-entraining flow in the wake of three-dimensional rectangular dry transom sterns with varying speeds and half-beam-to-draft ratios $B/D$. We employ two-phase (air/water), time-dependent simulations utilizing conservative volume-of-fluid (cVOF) and boundary data immersion (BDIM) methods to obtain the flow structure and large-scale air entrainment in the wake. We confirm that the convergent-corner-wave region that forms immediately aft of the stern wake is ballistic, thus predictable only by the speed and (rectangular) geometry of the ship. We show that the flow structure in the air–water mixed region contains a shear layer with a streamwise jet and secondary vortex structures due to the presence of the quasi-steady, three-dimensional breaking waves. We apply a Lagrangian cavity identification technique to quantify the air entrainment in the wake and show that the strongest entrainment is where wave breaking occurs. We identify an inverse dependence of the maximum average void fraction and total volume entrained with $B/D$. We determine that the average surface entrainment rate initially peaks at a location that scales with draft Froude number and that the normalized average air cavity density spectrum has a consistent value providing there is active air entrainment. A small parametric study of the rectangular geometry and stern speed establishes and confirms the scaling of the interface characteristics with draft Froude number and geometry. In Part 2 (Hendrikson & Yue, J. Fluid Mech., vol. 875, 2019, pp. 884–913) we examine the incompressible highly variable density turbulence characteristics and turbulence closure modelling.

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