The nonlinear behaviour of a constant vorticity layer at a wall

1981 ◽  
Vol 108 ◽  
pp. 401-421 ◽  
Author(s):  
D. I. Pullin
Keyword(s):  
Numerical Solutions ◽  
Free Stream ◽  
Outer Part ◽  
Single Mode ◽  
Nonlinear Behaviour ◽  
Material Interface ◽  
Constant Vorticity ◽  
Flow Features ◽  
Sinusoidal Disturbance ◽  
Finite Disturbance

The so-called ‘water-bag’ method is used to study the behaviour of a two-dimensional inviscid layer of constant vorticity ω and of mean thickness δ adjacent to a wall with slip at the wall. A nonlinear initial-value equation is derived which describes the motion of the material interface separating the rotational fluid within the layer from the irrotational free stream, for the case where this interface is subject to streamwise cyclic disturbances to its undisturbed shape. A linearized solution to this equation shows that a sinusoidal disturbance of wavelength λ propagates as one mode of a neutrally stable Kelvin-Helmholtz wave with velocity ωλ[1 − exp (−4πδ/λ)]/4π relative to the fluid at infinity. Numerical solutions of the full nonlinear equation for a range of wavelengths and finite disturbance amplitudes indicate different behaviour. For sufficiently large amplitude the interface valleys evolve into long re-entrant wedges of irrotational fluid which are ‘entrained’ into the layer and which are separated from the free stream by lobes or bulges of rotational fluid. This single-mode nonlinear interfacial distortion could be generated over a broad wavelength range with no indication of preferential scaling based on δ. It is suggested that the interface behaviour bears distinct resemblance to flow features observed at the interface between turbulent and non-turbulent fluid in recent smoke-in-air flow-visualization studies of the outer part of a constant pressure turbulent boundary layer. The calculated rotational fluid lobe velocities, which are not very different from the equivalent linearized wave velocities, are found to be in reasonable agreement with the few existing measurements of the velocity of bulges at the turbulent–nonturbulent fluid interface, while the computed velocity field in the lobe is in qualitative agreement with the general flow pattern observed in experiments. In the absence of a preferred scale or range of scales for the development of the interfacial distortion, however, it is concluded that the present results cannot be interpreted as supporting the hypothesis of the presence of largescale coherent motions in the outer part of the layer.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry

1989 ◽  
Vol 209 ◽  
pp. 285-308 ◽  
Author(s):  
R. J. Bodonyi ◽  
W. J. C. Welch ◽  
P. W. Duck ◽  
M. Tadjfar
Keyword(s):  
Numerical Solutions ◽  
Free Stream ◽  
Stokes Equations ◽  
Numerical Study ◽  
Navier Stokes ◽  
Surface Geometry ◽  
Navier Stokes Equations ◽  
S Waves ◽  
Tollmien Schlichting

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

Effects of Unsteady Free-Stream Velocity and Free-Stream Turbulence at a Stagnation Point

1983 ◽  
Vol 105 (1) ◽  
pp. 66-71 ◽  
Author(s):  
R. S. R. Gorla
Keyword(s):  
Stagnation Point ◽  
Numerical Solutions ◽  
Free Stream ◽  
Eddy Diffusivity ◽  
Descent Method ◽  
Free Stream Velocity ◽  
Steepest Descent Method ◽  
Wall Friction ◽  
Stream Velocity ◽  
Free Stream Turbulence

An analysis is presented to investigate the combined effects of transient free-stream velocity and free-stream turbulence at a stagnation point on a cylinder situated in a crossflow. A model has been successfully formulated for the eddy diffusivity induced by the free-stream turbulence. The governing momentum equation has been integrated by the steepest descent method. Numerical solutions are provided for the unsteady wall shear stress function for specific free-stream transients. The results are correlated by a new turbulence parameter. It has been found that the wall friction increases with increasing free-stream turbulence intensity. In the case of flows involving unsteady free-stream velocity, the friction factor increases with increasing values of the reduced frequency of oscillations.

Nonlinear behaviour of convergent Richtmyer–Meshkov instability

2019 ◽  
Vol 877 ◽  
pp. 130-141 ◽  
Author(s):  
Xisheng Luo ◽  
Ming Li ◽  
Juchun Ding ◽  
Zhigang Zhai ◽  
Ting Si
Keyword(s):  
Shock Tube ◽  
Linear Models ◽  
Early Stage ◽  
Soap Film ◽  
Single Mode ◽  
Planar Geometry ◽  
Nonlinear Behaviour ◽  
Strong Nonlinearity ◽  
Convex Wall ◽  
Wall Profile

A novel shock tube is designed to investigate the nonlinear feature of convergent Richtmyer–Meshkov instability on a single-mode interface formed by a soap film technique. The shock tube employs a concave–oblique–convex wall profile which first transforms a planar shock into a cylindrical arc, then gradually strengthens the cylindrical shock along the oblique wall, and finally converts it back into a planar one. Therefore, the new facility can realize analysis on compressibility and nonlinearity of convergent Richtmyer–Meshkov instability by eliminating the interface deceleration and reshock. Five sinusoidal $\text{air}{-}\text{SF}_{6}$ interfaces with different amplitudes and wavelengths are considered. For all cases, the perturbation amplitude experiences a linear growth much longer than that in the planar geometry. A compressible linear model is derived by considering a constant uniform fluid compression, which shows a slight difference to the incompressible theory. However, both the linear models overestimate the perturbation growth from a very early stage due to the presence of strong nonlinearity. The nonlinear model of Wang et al. (Phys. Plasmas, vol. 22, 2015, 082702) is demonstrated to predict well the amplitude growth up to a normalized time of 1.0. The prolongation of the linear increment is mainly ascribed to the counteraction between the promotion by geometric convergence and the suppression by nonlinearity. Growths of the first three harmonics, obtained by a Fourier analysis of the interface contour, provide a first thorough validation of the nonlinear theory.

scholarly journals The influence of vertical density and velocity distributions on snow avalanche runout

2010 ◽  
Vol 51 (54) ◽  
pp. 200-206 ◽  
Author(s):  
Ashok K. Keshari ◽  
Deba P. Satapathy ◽  
Amod Kumar
Keyword(s):  
Flux Distribution ◽  
Numerical Solutions ◽  
Model Parameters ◽  
Dynamics Model ◽  
Turbulent Friction ◽  
One Dimensional ◽  
Variation Diminishing ◽  
Flow Features ◽  
Vertical Density ◽  
Two Fluid

AbstractA one-dimensional avalanche dynamics model accounting for vertical density and velocity distributions is presented. Mass and momentum flux distribution factors are derived to incorporate the effect of density and velocity variations within the framework of depth-integrated models. Using experiments of avalanche flows on an inclined snow chute at Dhundhi, Manali, India, we conceptualize snow flow rheology as a Voellmy fluid where the distribution of internal shearing is given by a Newtonian fluid (NF) or Criminale–Ericksen–Filbey fluid (CEFF). Then the generalized mass and momentum distribution factors are computed for these two fluid models for different density stratifications. Numerical solutions are obtained using a total variation diminishing Lax–Friedrichs (TVDLF) finite-difference method. The model is validated with the experimental results. We find that the flow features of the chute experiments are simulated well by the model. The velocities and runout distances are obtained for the Voellmy model with both NF and CEFF extensions for various input volumes, and the optimum values of the model parameters, namely, coefficients of dynamic and turbulent friction, are determined.

Laminar flow past an abruptly accelerated elliptic cylinder at 45° incidence

1974 ◽  
Vol 65 (4) ◽  
pp. 711-734 ◽  
Author(s):  
H. J. Lugt ◽  
H. J. Haussling
Keyword(s):  
Steady State ◽  
Numerical Solutions ◽  
Fluid Flows ◽  
Elliptic Cylinder ◽  
Finite Difference Schemes ◽  
Quasi Steady State ◽  
Constant Vorticity ◽  
Transient Period ◽  
Incompressible Fluid Flows ◽  
Steady State Solutions

Numerical solutions for laminar incompressible fluid flows past an abruptly started elliptic cylinder at 45° incidence are presented. Various finite-difference schemes for the stream-function/vorticity formulation are used and their merits briefly discussed. Almost steady-state solutions are obtained forRe= 15 and 30, whereas forRe= 200 a Kármán vortex street develops. The transient period from the start to the steady or quasi-steady state is investigated in terms of patterns of streamlines and lines of constant vorticity and drag, lift and moment coefficients.

Unstart phenomena induced by mass addition and heat release in a model scramjet

2016 ◽  
Vol 797 ◽  
pp. 604-629 ◽  
Author(s):  
S. Im ◽  
D. Baccarella ◽  
B. McGann ◽  
Q. Liu ◽  
L. Wermer ◽  
...  
Keyword(s):  
Heat Release ◽  
Mass Flow Rate ◽  
Mass Flow ◽  
Flow Rate ◽  
High Speed ◽  
Free Stream ◽  
Mass Loading ◽  
High Enthalpy ◽  
Hypersonic Wind Tunnel ◽  
Flow Features

The unstart phenomena in a model scramjet with a free stream Mach number of 4.5 were investigated at an arc-heated hypersonic wind tunnel. High-speed schlieren imaging and high resonance frequency pressure measurements were used to capture the flow features during the unstart process. Three unstart conditions were tested: (i) a low-enthalpy free stream with mass loading, (ii) a high-enthalpy free stream with mass loading and (iii) a high-enthalpy free stream with mass loading and heat release. It was revealed that the unstart threshold and the time from the onset to the completion of unstart depended strongly on the mass loading rate and the heat exchange. The negative heat addition (cooling) significantly increased the threshold of mass flow rate triggering unstart. The decrement of the mass flow rate threshold for unstart was observed in the presence of heat release by combustion. The observed transient and quasi-steady behaviours of the unstart shockwave system and the jet motion were similar in all of the test conditions. On the other hand, at the lip of inlet model, the unstart shockwave under the cold free stream condition exhibited a relatively steady behaviour while severe oscillatory flow motions of the jet and the unstart shockwave were observed in the combustion-driven unstart process. The different unstarted flow behaviours between the three flow conditions were explained using a simplified one-dimensional flow choking analysis and use of the Korkegi criterion.

Transient Temperature Distributions in a Cylinder Heated by Microwaves

1996 ◽  
Vol 430 ◽  
Author(s):  
H. W. Jackson ◽  
M. Barmatz ◽  
P. Wagner
Keyword(s):  
Numerical Solutions ◽  
Single Mode ◽  
Dielectric Constants ◽  
Microwave Cavity ◽  
Transient Temperature ◽  
Hybrid Heating ◽  
Fine Mesh ◽  
Temperature Distributions ◽  
Analytic Expressions ◽  
Lossy Dielectric

AbstractTransient temperature distributions were calculated for a lossy dielectric cylinder coaxially aligned in a cylindrical microwave cavity excited in a single mode. Results were obtained for sample sizes that range from fibers to large cylinders. Realistic values for temperature dependent complex dielectric constants and thermophysical properties of the samples were used. Losses in cavity walls were taken into account as were realistic thermal emissivities at all surfaces. For a fine mesh of points in time, normal mode properties and microwave power absorption profiles were evaluated using analytic expressions. Those expressions correspond to exact solutions of Maxwell's equations within the framework of a cylindrical shell model. Heating produced by the microwave absorption was included in self-consistent numerical solutions of thermal equations. In this model, both direct microwave heating and radiant heating of the sample (hybrid heating) were studied by including a lossy dielectric tube surrounding the sample. Calculated results are discussed within the context of two parametric studies. One is concerned with relative merits of microwave and hybrid heating of fibers, rods, and larger cylinders. The other is concerned with thermal runaway.

scholarly journals Turbulent Transport Measurements in a Heated Boundary Layer: Combined Effects of Free-Stream Turbulence and Removal of Concave Curvature

1995 ◽  
Author(s):  
Michael D. Kestoras ◽  
Terrence W. Simon
Keyword(s):  
Boundary Layer ◽  
Turbulence Intensity ◽  
Free Stream ◽  
Turbulent Transport ◽  
Outer Part ◽  
Test Facility ◽  
Free Stream Turbulence ◽  
Concave Wall ◽  
Prandtl Numbers ◽  
Near Wall

Turbulence measurements for both momentum and heat transport are taken in a boundary layer over a flat, recovery wall downstream of a concave wall (R=0.97m). The boundary layer appears turbulent from the beginning of the concave wall and grows over the test wall with negligible streamwise acceleration. The strength of curvature at the bend exit, δ99.5/R, is 0.04. The free-stream turbulence intensity is ∼8% at the beginning of the curve and is nearly uniform at ∼4.5% throughout the recovery wall. Comparisons are made with data taken in an earlier study, in the same test facility, but with a low free-stream turbulence intensity (−0.6%). Results show that on the recovery wall, elevated free-stream turbulence intensity enhances turbulent transport quantities such as -uv¯ and vt¯ in most of the outer part of the boundary layer, but near-wall values of vt¯ remain unaffected. This is in contrast to near-wall vt¯ values within the curve which decrease when free-stream turbulence is increased. At the bend exit, decreases of -uv¯ and vt¯ due to removal of curvature become more profound when free-stream turbulence intensity is elevated, compared to low-TI behavior. Measurements in the core of the flow indicate that the high levels of cross transport of momentum over the upstream concave wall cease when curvature is removed. Other results show that turbulent Prandtl numbers over the recovery wall are reduced to −0.9 when free-stream turbulence intensity is elevated, consistent with the rise in Stanton numbers over the recovery wall.

NUMERICAL SOLUTION OF SINGLE MODE GYROTRON EQUATION

2005 ◽  
Vol 9 (1) ◽  
pp. 25-38 ◽  
Author(s):  
O. Dumbrajs ◽  
H. Kalis ◽  
A. Reinfelds
Keyword(s):  
Numerical Solution ◽  
Finite Differences ◽  
Numerical Solutions ◽  
Single Mode ◽  
Eigenvalues And Eigenfunctions ◽  
Analytical And Numerical Solutions ◽  
Quasistationary Solutions

In this paper we study numerical problems arising in solving the single mode gyrotron equation. Using the method of finite differences analytical and numerical solutions are obtained. Quasistationary solutions and corresponding eigenvalues and eigenfunctions of this problem are investigated.

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