An Investigation Into Nonlinear Growth Rate of Two-Dimensional and Three-Dimensional Single-Mode Richtmyer–Meshkov Instability Using an Arbitrary-Lagrangian–Eulerian Algorithm

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Mike Probyn ◽  
Ben Thornber ◽  
Dimitris Drikakis ◽  
David Youngs ◽  
Robin Williams
Keyword(s):  
Growth Rate ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Numerical Study ◽  
Three Dimensional ◽  
Single Mode ◽  
Computational Time ◽  
Initial Growth ◽  
Arbitrary Lagrangian Eulerian ◽  
Layer Width

This paper presents an investigation into the use of a moving mesh algorithm for solving unsteady turbulent mixing problems. The growth of a shock induced mixing zone following reshock, using an initial setup comparable to that of existing experimental work, is used to evaluate the behavior of the numerical scheme for single-mode Richtmyer–Meshkov instability (SM-RMI). Subsequently the code is used to evaluate the growth rate for a range of different initial conditions. The initial growth rate for three-dimensional (3D) SM Richtmyer–Meshkov is also presented for a number of different initial conditions. This numerical study details the development of the mixing layer width both prior to and after reshock. The numerical scheme used includes an arbitrary Lagrangian–Eulerian grid motion which is successfully used to reduce the mesh size and computational time while retaining the accuracy of the simulation results. Varying initial conditions shows that the growth rate after reshock is independent of the initial conditions for a SM provided that the initial growth remains in the linear regime.

The Richtmyer–Meshkov instability of a ‘V’ shaped air/ interface

2016 ◽  
Vol 802 ◽  
pp. 186-202 ◽  
Author(s):  
Xisheng Luo ◽  
Ping Dong ◽  
Ting Si ◽  
Zhigang Zhai
Keyword(s):  
Growth Rate ◽  
High Speed ◽  
Initial Conditions ◽  
Monotone Function ◽  
Flow Characteristics ◽  
Soap Film ◽  
Vertex Angle ◽  
Initial Growth ◽  
General Behaviour ◽  
Interface Width

The Richtmyer–Meshkov instability on a ‘V’ shaped air/SF$_{6}$ gaseous interface is experimentally studied in a shock tube. By the soap film technique, a discontinuous interface without supporting mesh is formed so that the initial conditions of the interface can be accurately controlled. Five ‘V’ shaped air/$\text{SF}_{6}$ interfaces with different vertex angles ($60^{\circ }$, $90^{\circ }$, $120^{\circ }$, $140^{\circ }$ and $160^{\circ }$) are created where the ratio of the initial interface amplitude to the wavelength varies to highlight the effects of initial condition on the flow characteristics. The wave patterns and interface morphologies are clearly identified in the high-speed schlieren sequences, which show that the interface deforms in a less pronounced manner with less vortices generated as the vertex angle increases. A regime change is observed in the interface width growth rate near a vertex angle of $160^{\circ }$, which provides an experimental evidence for the numerical results obtained by McFarland et al. (Phys. Scr. vol. T155, 2013, 014014). The growth rate of interface width in the linear phase is compared with the theoretical predictions from the classical impulsive model and a modified linear model, and the latter is proven to be effective for a moderate to large initial amplitude. It is found that the initial growth rate of the interface width is a non-monotone function of the initial vertex angle (amplitude–wavelength ratio), i.e. the interface width growth rate in the linear stage experiences an increase and then a decrease as the vertex angle increases. A similar conclusion was also reached by Dell et al. (Phys. Plasmas, vol. 22, 2015, 092711) numerically for a sinusoidal interface. Finally, the general behaviour of the interface width growth in the nonlinear stage can be well captured by the nonlinear model proposed by Dimonte & Ramaprabhu (Phys. Fluids, vol. 22, 2010, 014104).

Solving linear systems from dynamical energy analysis - using and reusing preconditioners

2021 ◽  
Vol 263 (5) ◽  
pp. 1041-1052
Author(s):  
Martin Richter ◽  
Gregor Tanner ◽  
Bruno Carpentieri ◽  
David J. Chappell
Keyword(s):  
Wave Equations ◽  
Energy Analysis ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Iterative Solvers ◽  
Computational Method ◽  
Computational Time ◽  
Shell Elements ◽  
Finite Dimensional ◽  
Large Matrix

Dynamical energy analysis (DEA) is a computational method to address high-frequency vibro-acoustics in terms of ray densities. It has been used to describe wave equations governing structure-borne sound in two-dimensional shell elements as well as three-dimensional electrodynamics. To describe either of those problems, the wave equation is reformulated as a propagation of boundary densities. These densities are expressed by finite dimensional approximations. All use-cases have in common that they describe the resulting linear problem using a very large matrix which is block-sparse, often real-valued, but non-symmetric. In order to efficiently use DEA, it is therefore important to also address the performance of solving the corresponding linear system. We will cover three aspects in order to reduce the computational time: The use of preconditioners, properly chosen initial conditions, and choice of iterative solvers. Especially the aspect of potentially reusing preconditioners for different input parameters is investigated.

Nonlinear cellular motions in Poiseuille channel flow

1974 ◽  
Vol 64 (2) ◽  
pp. 319-346 ◽  
Author(s):  
J.-P. Zahn ◽  
Juri Toomre ◽  
E. A. Spiegel ◽  
D. O. Gough
Keyword(s):  
Fourier Series ◽  
Steady State ◽  
Infinite System ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Single Mode ◽  
Frame Of Reference ◽  
Moving Frame ◽  
System Of Equations ◽  
Plane Poiseuille Flow

We expand the equations describing plane Poiseuille flow in Fourier series in the co-ordinates in the plane parallel to the bounding walls. There results an infinite system of equations for the amplitudes, which are functions of time and of the cross-stream co-ordinate. This system is drastically truncated and the resulting set of equations is solved accurately by a finite difference method. Three truncations are considered: (I) a single mode with dependence only on the downstream co-ordinate and time, (II) the mode of (I) plus its first harmonic, (III) a single three-dimensional mode. For all three cases, for a variety of initial conditions, the solutions evolve to a steady state as seen in a particular moving frame of reference. No runaways are encountered.

Fabrication of GaN Films in a Chemical Vapor Deposition Reactor

2013 ◽  
Author(s):  
J. Meng ◽  
Y. Jaluria ◽  
S. Wong
Keyword(s):  
Growth Rate ◽  
Film Growth ◽  
Numerical Study ◽  
Three Dimensional ◽  
Chemical Vapor ◽  
Rotating Disk ◽  
Transport Processes ◽  
Operating Conditions ◽  
Fluid Loss ◽  
Flow Recirculation

A three-dimensional numerical study has been carried out on the rotating disk GaN MOCVD process, and it is also coupled with an experimental study on the flow and thermal transport processes in the system. An impingement type reactor, with a rotating base, is considered. The dependence of the thin film growth rate and uniformity on operating conditions such as inflow velocity, rotational speed, and susceptor temperature are investigated in detail. Similarly, the effect of the geometry and configuration of the reactor are studied. The study also considers the effect of thermal and solutal buoyancy on the resulting flow. The flow and the associated transport processes are discussed in detail on the basis of the results obtained to suggest approaches to improve the uniformity of the film, minimize fluid loss and reduce flow recirculation that could affect growth rate and uniformity.

An Approximate Three-Dimensional Solution for Melting or Freezing Around a Buried Pipe Beneath a Free Surface

1986 ◽  
Vol 108 (4) ◽  
pp. 900-906 ◽  
Author(s):  
G.-P. Zhang ◽  
S. Weinbaum ◽  
L. M. Jiji
Keyword(s):  
Approximate Solution ◽  
Free Surface ◽  
Phase Change ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Computational Time ◽  
Buried Pipe ◽  
Dimensional Solution ◽  
Pipe Surface

This paper presents a quasi-steady-state approximate solution for small Stefan number for the three-dimensional melting or freezing around a fluid-carrying pipe buried in a semi-infinite phase change medium (PCM). The two-dimensional quasi-steady approximate solution method, the virtual free surface technique [18], has been extended to three dimensions where axial thermal interaction between the moving fluid and the PCM is considered. Of particular interest in the motion of the phase change interface and the time variation of the axial temperature distribution in the fluid. Due to the singularities of the differential equations along the pipe surface, an axisymmetric analytic solution is provided for the region near the pipe wall. Solutions are presented for several representative dimensionless pipe burial depths and initial conditions. The computational time to predict the three-dimensional interface location up to 10 years is several minutes on an IBM 4341 computer.

scholarly journals Stabilizing effect of optimally amplified streaks in parallel wakes

2013 ◽  
Vol 739 ◽  
pp. 37-56 ◽  
Author(s):  
Gerardo Del Guercio ◽  
Carlo Cossu ◽  
Gregory Pujals
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Initial Conditions ◽  
Initial Perturbation ◽  
Three Dimensional ◽  
Maximum Amplitude ◽  
Maximum Growth ◽  
Absolute Instability ◽  
Stabilizing Effect ◽  
The Absolute

AbstractWe show that optimal perturbations artificially forced in parallel wakes can be used to completely suppress the absolute instability and to reduce the maximum temporal growth rate of the inflectional instability. To this end we compute optimal transient energy growths of stable streamwise uniform perturbations supported by a parallel wake for a set of Reynolds numbers and spanwise wavenumbers. The maximum growth rates are shown to be proportional to the square of the Reynolds number and to increase with spanwise wavelengths with sinuous perturbations slightly more amplified than varicose ones. Optimal initial conditions consist of streamwise vortices and the optimally amplified perturbations are streamwise streaks. Families of nonlinear streaky wakes are then computed by direct numerical simulation using optimal initial vortices of increasing amplitude as initial conditions. The stabilizing effect of nonlinear streaks on temporal and spatiotemporal growth rates is then determined by analysing the linear impulse response supported by the maximum amplitude streaky wakes profiles. This analysis reveals that at $\mathit{Re}= 50$, streaks of spanwise amplitude ${A}_{s} \approx 8\hspace{0.167em} \% {U}_{\infty } $ can completely suppress the absolute instability, converting it into a convective instability. The sensitivity of the absolute and maximum temporal growth rates to streak amplitudes is found to be quadratic, as has been recently predicted. As the sensitivity to two-dimensional (2D, spanwise uniform) perturbations is linear, three-dimensional (3D) perturbations become more effective than the 2D ones only at finite amplitudes. Concerning the investigated cases, 3D perturbations become more effective than the 2D ones for streak amplitudes ${A}_{s} \gtrsim 3\hspace{0.167em} \% {U}_{\infty } $ in reducing the maximum temporal amplification and ${A}_{s} \gtrsim 12\hspace{0.167em} \% {U}_{\infty } $ in reducing the absolute growth rate. However, due to the large optimal energy growths they experience, 3D optimal perturbations are found to be much more efficient than 2D perturbations in terms of initial perturbation amplitudes. Despite their lower maximum transient amplification, varicose streaks are found to be always more effective than sinuous ones in stabilizing the wakes, in accordance with previous findings.

scholarly journals Growth rate of a shocked mixing layer with known initial perturbations

2013 ◽  
Vol 725 ◽  
pp. 372-401 ◽  
Author(s):  
Christopher R. Weber ◽  
Andrew W. Cook ◽  
Riccardo Bonazza
Keyword(s):  
Growth Rate ◽  
Mass Flux ◽  
Initial Conditions ◽  
Mixing Layer ◽  
Streamwise Velocity ◽  
Initial Growth ◽  
Initial Growth Rate ◽  
Baroclinic Torque ◽  
Shock Density ◽  
Post Shock

AbstractWe derive a growth-rate model for the Richtmyer–Meshkov mixing layer, given arbitrary but known initial conditions. The initial growth rate is determined by the net mass flux through the centre plane of the perturbed interface immediately after shock passage. The net mass flux is determined by the correlation between the post-shock density and streamwise velocity. The post-shock density field is computed from the known initial perturbations and the shock jump conditions. The streamwise velocity is computed via Biot–Savart integration of the vorticity field. The vorticity deposited by the shock is obtained from the baroclinic torque with an impulsive acceleration. Using the initial growth rate and characteristic perturbation wavelength as scaling factors, the model collapses the growth-rate curves and, in most cases, predicts the peak growth rate over a range of Mach numbers ($1. 1\leq {M}_{i} \leq 1. 9$), Atwood numbers ($- 0. 73\leq A\leq - 0. 35$ and $0. 22\leq A\leq 0. 73$), adiabatic indices ($1. 40/ 1. 67\leq {\gamma }_{1} / {\gamma }_{2} \leq 1. 67/ 1. 09$) and narrow-band perturbation spectra. The mixing layer at late times exhibits a power-law growth with an average exponent of $\theta = 0. 24$.

Combined Experimental and Numerical Study for Multiple Microchannel Heat Transfer System

2010 ◽  
Author(s):  
Jingru Zhang ◽  
Yogesh Jaluria ◽  
Tiantian Zhang ◽  
Li Jia
Keyword(s):  
Experimental Data ◽  
Numerical Simulation ◽  
Steady State ◽  
Numerical Model ◽  
Initial Conditions ◽  
Numerical Study ◽  
Numerical Models ◽  
Three Dimensional ◽  
Variation Versus ◽  
Heat Transfer System

Multiple microchannel heat sinks for potential use for electronic chip cooling are studied experimentally and numerically to characterize their thermal performance. The numerical simulation is driven by experimental data, which are obtained concurrently, to obtain realistic, accurate and validated numerical models. The ultimate goal is to design and optimize thermal systems. The experimental setup was established and liquid flow in the multiple microchannels was studied under different flow rates and heat influx. The temperature variation versus time was recorded by thermocouples, from which the time needed to reach steady state was determined. Temperature variations under steady state conditions were compared with three-dimensional steady state numerical simulation for the same boundary and initial conditions. The experimental data served as input parameters for the validation of the numerical model. In case of discrepancy, the numerical model was improved. A fairly good agreement between the experimental and simulation results was obtained. The numerical model also served to provide input that could be employed to improve and modify the experimental arrangement.

scholarly journals Late-time growth of the Richtmyer–Meshkov instability for different Atwood numbers and different dimensionalities

2003 ◽  
Vol 21 (3) ◽  
pp. 363-368 ◽  
Author(s):  
A. YOSEF-HAI ◽  
O. SADOT ◽  
D. KARTOON ◽  
D. ORON ◽  
L.A. LEVIN ◽  
...  
Keyword(s):  
Growth Rate ◽  
Theoretical Model ◽  
Numerical Simulations ◽  
Initial Perturbation ◽  
Three Dimensional ◽  
Single Mode ◽  
Late Time ◽  
Two Dimensional ◽  
Good Agreement

The late-time growth rate of the Richtmyer–Meshkov instability was experimentally studied at different Atwood numbers with two-dimensional (2D) and three-dimensional (3D) single-mode initial perturbations. The results of these experiments were found to be in good agreement with the results of the theoretical model and numerical simulations. In another set of experiments a bubble-competition phenomenon, which was observed in previous work for 2D initial perturbation (Sadotet al., 1998), was shown to exist also when the initial perturbation is of a 3D nature.

A Numerical Study of Three-Dimensional Roll Cells within Rigid Boundaries

1979 ◽  
Vol 101 (2) ◽  
pp. 233-237 ◽  
Author(s):  
A. M. C. Chan ◽  
S. Banerjee
Keyword(s):  
Natural Convection ◽  
Initial Conditions ◽  
Numerical Study ◽  
Numerical Technique ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Thermal Boundary Conditions ◽  
Aspect Ratios ◽  
Convection Roll ◽  
Marker And Cell Method

Three-dimensional natural convection, roll cells within rigid enclosures have been studied with a previously developed numerical technique based on the marker and cell method. Given identical initial conditions, the velocity and temperatures fields are found to be sensitive to the thermal boundary conditions on the side walls and the aspect ratio of the enclosure. Two-dimensional results are also obtained and compared with the corresponding three-dimensional results. The two-dimensional calculations do not agree well with the three dimensional ones, especially for enclosures having aspect ratios less than unity. This indicates that care must be taken in analyzing natural convection problems of this type with two-dimensional methods.

Sign in / Sign up

Export Citation Format

Share Document