The Simulation of Mixing Layers Driven by Compound Buoyancy and Shear

1996 ◽  
Vol 118 (2) ◽  
pp. 370-376 ◽  
Author(s):  
D. M. Snider ◽  
M. J. Andrews
Keyword(s):  
Growth Rate ◽  
Flow Field ◽  
Numerical Solution ◽  
Mixing Layer ◽  
Measured Data ◽  
Mixing Layers ◽  
Two Dimensional ◽  
One Dimensional ◽  
Self Similar ◽  
Accurate Numerical Solution

Fully developed compound shear and buoyancy driven mixing layers are predicted using a k-ε turbulence model. Such mixing layers present an exchange of equilibrium in mixing flows. The k-ε buoyancy constant Cε3 = 0.91, defined in this study for buoyancy unstable mixing layers, is based on an approximate self-similar analysis and an accurate numerical solution. One-dimensional transient and two-dimensional steady calculations are presented for buoyancy driven mixing in a uniform flow field. Two-dimensional steady calculations are presented for compound shear and buoyancy driven mixing. The computed results for buoyancy alone and compound shear and buoyancy mixing compare well with measured data. Adding shear to an unstable buoyancy mixing layer does not increase the mixing growth rate compared with that from buoyancy alone. The nonmechanistic k-ε model which balances energy generation and dissipation using constants from canonical shear and buoyancy studies predicts the suppression of the compound mixing width. Experimental observations suggest that a reduction in growth rate results from unequal stream velocities that skew and stretch the normally vertical buoyancy plumes producing a reduced mixing envelope width.

scholarly journals Growth Rate Exponents of Richtmyer-Meshkov Mixing Layers

2005 ◽  
Vol 73 (3) ◽  
pp. 461-468 ◽  
Author(s):  
Timothy T. Clark ◽  
Ye Zhou
Keyword(s):  
Flow Field ◽  
Power Law ◽  
Mixing Layer ◽  
Power Laws ◽  
Mixing Layers ◽  
Spatial Gradients ◽  
Power Law Exponent ◽  
Virtual Time ◽  
Time Origin ◽  
Density Ratios

The Richtmyer-Meshkov mixing layer is initiated by the passing of a shock over an interface between fluid of differing densities. The energy deposited during the shock passage undergoes a relaxation process during which the fluctuational energy in the flow field decays and the spatial gradients of the flow field decrease in time. This late stage of Richtmyer-Meshkov mixing layers is studied from the viewpoint of self-similarity. Analogies with weakly anisotropic turbulence suggest that both the bubble-side and spike-side widths of the mixing layer should evolve as power-laws in time, with the same power-law exponent and virtual time origin for both sides. The analogy also bounds the power-law exponent between 2∕7 and 1∕2. It is then shown that the assumption of identical power-law exponents for bubbles and spikes yields fits that are in good agreement with experiment at modest density ratios.

Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet

2016 ◽  
Vol 14 (01) ◽  
pp. 1650004 ◽  
Author(s):  
M. Tahami ◽  
A. Askari Hemmat ◽  
S. A. Yousefi
Keyword(s):  
Integral Equation ◽  
Tensor Product ◽  
Numerical Solution ◽  
Fredholm Integral Equation ◽  
Fredholm Integral Equations ◽  
Wavelet Basis ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
One Dimensional

In one-dimensional problems, the Legendre wavelets are good candidates for approximation. In this paper, we present a numerical method for solving two-dimensional first kind Fredholm integral equation. The method is based upon two-dimensional linear Legendre wavelet basis approximation. By applying tensor product of one-dimensional linear Legendre wavelet we construct a two-dimensional wavelet. Finally, we give some numerical examples.

Compressibility effects in a turbulent annular mixing layer. Part 1. Turbulence and growth rate

2000 ◽  
Vol 421 ◽  
pp. 229-267 ◽  
Author(s):  
JONATHAN B. FREUND ◽  
SANJIVA K. LELE ◽  
PARVIZ MOIN
Keyword(s):  
Growth Rate ◽  
Mach Number ◽  
Mixing Layer ◽  
Mixing Layers ◽  
Velocity Difference ◽  
Mean Velocity ◽  
The Mean ◽  
Mach Numbers ◽  
High Mach ◽  
Compressibility Effects

This work uses direct numerical simulations of time evolving annular mixing layers, which correspond to the early development of round jets, to study compressibility effects on turbulence in free shear flows. Nine cases were considered with convective Mach numbers ranging from Mc = 0.1 to 1.8 and turbulence Mach numbers reaching as high as Mt = 0.8.Growth rates of the simulated mixing layers are suppressed with increasing Mach number as observed experimentally. Also in accord with experiments, the mean velocity difference across the layer is found to be inadequate for scaling most turbulence statistics. An alternative scaling based on the mean velocity difference across a typical large eddy, whose dimension is determined by two-point spatial correlations, is proposed and validated. Analysis of the budget of the streamwise component of Reynolds stress shows how the new scaling is linked to the observed growth rate suppression. Dilatational contributions to the budget of turbulent kinetic energy are found to increase rapidly with Mach number, but remain small even at Mc = 1.8 despite the fact that shocklets are found at high Mach numbers. Flow visualizations show that at low Mach numbers the mixing region is dominated by large azimuthally correlated rollers whereas at high Mach numbers the flow is dominated by small streamwise oriented structures. An acoustic timescale limitation for supersonically deforming eddies is found to be consistent with the observations and scalings and is offered as a possible explanation for the decrease in transverse lengthscale.

The compressible vortex pair

1990 ◽  
Vol 220 ◽  
pp. 339-354 ◽  
Author(s):  
S. D. Heister ◽  
J. M. Mcdonough ◽  
A. R. Karagozian ◽  
D. W. Jenkins
Keyword(s):  
Flow Field ◽  
Numerical Solution ◽  
Numerical Method ◽  
Compressible Flow ◽  
Vortex Pair ◽  
Flow Speed ◽  
Special Treatment ◽  
Two Dimensional ◽  
Potential Equation ◽  
Weak Shocks

A numerical solution for the flow field associated with a compressible pair of counter-rotating vortices is developed. The compressible, two-dimensional potential equation is solved utilizing the numerical method of Osher et al. (1985) for flow regions in which a non-zero density exists. Close to the vortex centres, vacuum ‘cores’ develop owing to the existence of a maximum achievable flow speed in a compressible flow field. A special treatment is required to represent these vacuum cores. Typical streamline patterns and core boundaries are obtained for upstream Mach numbers as high as 0.3, and the formation of weak shocks, predicted by Moore & Pullin (1987), is observed.

scholarly journals A numerical study of a variable-density low-speed turbulent mixing layer

2017 ◽  
Vol 830 ◽  
pp. 569-601 ◽  
Author(s):  
Antonio Almagro ◽  
Manuel García-Villalba ◽  
Oscar Flores
Keyword(s):  
Growth Rate ◽  
Mach Number ◽  
Incompressible Flows ◽  
Numerical Study ◽  
Density Ratio ◽  
Mixing Layer ◽  
Variable Density ◽  
The Self ◽  
Low Speed ◽  
Self Similar

Direct numerical simulations of a temporally developing, low-speed, variable-density, turbulent, plane mixing layer are performed. The Navier–Stokes equations in the low-Mach-number approximation are solved using a novel algorithm based on an extended version of the velocity–vorticity formulation used by Kim et al. (J. Fluid Mech., vol 177, 1987, 133–166) for incompressible flows. Four cases with density ratios $s=1,2,4$ and 8 are considered. The simulations are run with a Prandtl number of 0.7, and achieve a $Re_{\unicode[STIX]{x1D706}}$ up to 150 during the self-similar evolution of the mixing layer. It is found that the growth rate of the mixing layer decreases with increasing density ratio, in agreement with theoretical models of this phenomenon. Comparison with high-speed data shows that the reduction of the growth rates with increasing density ratio has a weak dependence with the Mach number. In addition, the shifting of the mixing layer to the low-density stream has been characterized by analysing one-point statistics within the self-similar interval. This shifting has been quantified, and related to the growth rate of the mixing layer under the assumption that the shape of the mean velocity and density profiles do not change with the density ratio. This leads to a predictive model for the reduction of the growth rate of the momentum thickness, which agrees reasonably well with the available data. Finally, the effect of the density ratio on the turbulent structure has been analysed using flow visualizations and spectra. It is found that with increasing density ratio the longest scales in the high-density side are gradually inhibited. A gradual reduction of the energy in small scales with increasing density ratio is also observed.

Secondary instability of a temporally growing mixing layer

1987 ◽  
Vol 184 ◽  
pp. 207-243 ◽  
Author(s):  
Ralph W. Metcalfe ◽  
Steven A. Orszag ◽  
Marc E. Brachet ◽  
Suresh Menon ◽  
James J. Riley
Keyword(s):  
Growth Rates ◽  
Stokes Equations ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Mixing Layers ◽  
Navier Stokes ◽  
Small Scale ◽  
Two Dimensional ◽  
Mean Flow ◽  
Direct Numerical Integration

The three-dimensional stability of two-dimensional vortical states of planar mixing layers is studied by direct numerical integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective timescales, extract their energy from the mean flow and exist at moderately low Reynolds numbers. Their growth rates are comparable with the most rapidly growing inviscid instability and with the growth rates of two-dimensional subharmonic (pairing) modes. At high amplitudes, they can evolve into pairs of counter-rotating, streamwise vortices, connecting the primary spanwise vortices, which are very similar to the structures observed in laboratory experiments. The three-dimensional modes do not appear to saturate in quasi-steady states as do the purely two-dimensional fundamental and subharmonic modes in the absence of pairing. The subsequent evolution of the flow depends on the relative amplitudes of the pairing modes. Persistent pairings can inhibit three-dimensional instability and, hence, keep the flow predominantly two-dimensional. Conversely, suppression of the pairing process can drive the three-dimensional modes to more chaotic, turbulent-like states. An analysis of high-resolution simulations of fully turbulent mixing layers confirms the existence of rib-like structures and that their coherence depends strongly on the presence of the two-dimensional pairing modes.

scholarly journals Numerical solution of two-dimensional nonlinear fractional order reaction-advection-diffusion equation by using collocation method

2021 ◽  
Vol 29 (2) ◽  
pp. 211-230
Author(s):  
Manpal Singh ◽  
S. Das ◽  
Rajeev ◽  
E-M. Craciun
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Operational Matrix ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Existing Problems ◽  
Advection Diffusion Equation ◽  
Error Analyses ◽  
Nonlinear Algebraic Equations ◽  
Accurate Numerical Solution

Abstract In this article, two-dimensional nonlinear and multi-term time fractional diffusion equations are solved numerically by collocation method, which is used with the help of Lucas operational matrix. In the proposed method solutions of the problems are expressed in terms of Lucas polynomial as basis function. To determine the unknowns, the residual, initial and boundary conditions are collocated at the chosen points, which produce a system of nonlinear algebraic equations those have been solved numerically. The concerned method provides the highly accurate numerical solution. The accuracy of the approximate solution of the problem can be increased by expanding the terms of the polynomial. The accuracy and efficiency of the concerned method have been authenticated through the error analyses with some existing problems whose solutions are already known.

A Simple Model for Fluid Flow and Particle Motion Inside the Human Larynx

Volume 4 ◽  
2004 ◽  
Author(s):  
C. Ersahin ◽  
I. B. Celik ◽  
O. C. Elci ◽  
I. Yavuz ◽  
J. Li ◽  
...  
Keyword(s):  
Flow Field ◽  
Particle Deposition ◽  
Flow Distribution ◽  
Axial Direction ◽  
Correction Method ◽  
Accurate Solution ◽  
Two Dimensional ◽  
Transient Model ◽  
One Dimensional ◽  
Dimensional Flow

This study aims to develop a simple and quick, but sufficiently accurate solution method for calculating the air flow and tracking the particles in a complex tubular system, where the flow changes its magnitude and direction in a periodic manner. The flow field is assumed to be quasi-two-dimensional and a pressure-correction method is employed to calculate the spetio-temporal variation of the air velocity inside the larynx. Then, the calculated one-dimensional flow distribution is used to reconstruct a two-dimensional flow field is constructed based on the average velocity along the axial direction. The system geometry is taken as close as possible to the actual larynx for an average person with an average glottis opening. For the current study the walls of the larynx is approximated as rigid walls, but different ways to account for compliant walls are proposed within the context of the one-dimensional mode. The 1-D transient model is validated against a two-dimensional model using a verified commercial code. Particles are introduced into the system and tracked during every time fraction of the respiratory cycle. Then, the histograms of particles that come into contact with the larynx are calculated, and regions with a higher probability for particle deposition are identified.

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