Self-similar Flow Research Articles

On the theory of supersonic inviscid flow separation in gasdynamic problems

A general schematic flow representation that explains the mechanism of inviscid gas separation in time-dependent and three-dimensional gas flows is presented. The scenario of gas flow separation from a body surface or a mixing layer is described as a vortex which induces in the flowfield a velocity opposing to that of the main flow, thus decelerating it. Within the framework of this scenario the analytical conditions of separation are obtained for conical and self-similar gas flows which coincide with the results of experimental and numerical simulations.

Transonic shock and rarefaction wave interactions of two-dimensional Riemann problems for the self-similar nonlinear wave system

This paper addresses the self-similar transonic irrotational flow in gas dynamics in two space dimensions.We consider a configuration that the incident shock becomes a transonic shock as it enters the sonic circle, interacts with the rarefaction wave downstream, and then becomes sonic. The rarefaction wave further downstream becomes sonic (degenerate) creating an unknown boundary for the governing system. We present the Riemann data for this configuration, provide the characteristic decomposition of the system, and formulate the boundary value problem for this configuration. The numerical results are presented, and a method to establish the existence result is briefly discussed.

Unsteady self-similar viscous flow near a stagnation point

The problem of unsteady viscous incompressible fluid flow near a stagnation point is considered. Self-similar solutions describing plane and axisymmetric flows are constructed.

Challenging Mix Models on Transients to Self-Similarity of Unstably Stratified Homogeneous Turbulence

The present work aims at expanding the set of buoyancy-driven unstable reference flows—a critical ingredient in the development of turbulence models—by considering the recently introduced “Unstably Stratified Homogeneous Turbulence” (USHT) in both its self-similar and transient regimes. The previously established accuracy of an anisotropic Eddy-Damped Quasi-Normal Markovian Model (EDQNM) on the USHT has allowed us to: (i) build a data set of well defined transient flows from Homogeneous Isotropic Turbulence (HIT) to late-time self-similar USHT and (ii) on this basis, calibrate, validate, and compare three common Reynolds-Averaged Navier–Stokes (RANS) mixing models (two-equation, Reynolds stress, and two-fluid). The model calibrations were performed on the self-similar flows constrained by predefined long range correlations (Saffman or Batchelor type). Then, with fixed constants, validations were carried out over the various transients defined by the initial Froude number and mixing intensity. Significant differences between the models are observed, but none of them can accurately capture all of the transient regimes at once. Closer inspection of the various model responses hints at possible routes for their improvement.

Numerical simulation of self-similar thermal convection from a spinning cone in anisotropic porous medium

Self-similar steady natural convection thermal boundary layer flow from a rotating vertical cone to anisotropic Darcian porous medium is investigated theoretically and numerically. The transformed non-dimensional two-point boundary value problem is reduced to a system of coupled, highly nonlinear ordinary differential equations, which are solved subject to robust surface and free stream boundary conditions with the MAPLE 17 numerical quadrature software. Validation with earlier non-rotating studies is included, and also further verification of rotating solutions is achieved with a variational finite element method (FEM). The rotational (spin) parameter emerges as an inverse function of the Grashof number. The influence of this parameter, primary Darcy number, secondary Darcy number and Prandtl number on tangential velocity and swirl velocity, temperature and heat transfer rate are studied in detail. It is found that the dimensionless tangential velocity increases whilst the dimensionless swirl velocity and temperature decrease with the swirl Darcy number, tangential Darcy number and the rotational parameters. The model finds applications in chemical engineering filtration processing, liquid coating and spinning cone distillation columns.

Cratering of a solid body due to liquid impact

An impact between a high-speed liquid and a solid body leading to cratering during the impact is considered. The problem formulation is presented within the framework of a self-similar velocity potential flow for a liquid wedge impacting the solid wall. A linear constitutive relation is adopted, which links the speed of cratering with the pressure and shear stresses. The solution procedure is based on the integral hodograph method. It converts the boundary-value problem for the complex potential in the fluid domain into a system of integral equations which are solved using the method of successive approximations. The obtained solution is used to investigate the cratering mechanics during the impact. The results are presented in terms of the interface shape, the free surface shape, streamline patterns and pressure distribution along the interface.

Unsteady Self-Similar Viscous Flow Near a Critical Point

Unsteady Self-Similar Viscous Flow Near a Critical Point

The scaling of attention networks

We use clicks as a proxy of collective attention and construct networks to study the temporal dynamics of attention. In particular we collect the browsing records of millions of users on 1000 Web forums in two months. In the constructed networks, nodes are threads and edges represent the switch of users between threads in an hour. The investigated network properties include the number of threads N, the number of users UV, and the number of clicks, PV. We find scaling functions PV∼UVθ1, PV∼Nθ3, and UV∼Nθ2, in which the scaling exponents are always greater than 1. This means that (1) the studied networks maintain a self-similar flow structure in time, i.e., large networks are simply the scale-up versions of small networks; and (2) large networks are more “productive”, in the sense that an average user would generate more clicks in the larger systems. We propose a revised version of Zipf’s law to quantify the time-invariant flow structure of attention networks and relate it to the observed scaling properties. We also demonstrate the applied consequences of our research: forum-classification based on scaling properties.

Insights derived from CFD studies on the evolution of planar wall jets

ABSTRACTThe main objective of this paper is to demonstrate that a simple and cost-effective 2D Reynolds-averaged Navier–Stokes (RANS) simulation approach can often be efficiently used in industrial design applications. We simplified the designing approach of a racetrack jet dryer to a problem involving the streamwise evolution of an offset wall jet. We compared our simple 2D RANS simulations with experiments and large eddy simulation (LES), and were encouraged to see that our simplified approach produced a better correlation with the experiments compared to LES, which is expected to be much more accurate (even though computationally orders of magnitude more expensive). We conclude that under certain circumstances a simple 2D approach can lead to a dependable solution. Additionally, we used the results from these simulations to enhance our understanding of the evolution of offset wall jets. The insights derived from these simulations suggest the existence of scaling parameters that can express offset wall jets as a family of self-similar flows.

COMPARISON BETWEEN RHD SIMULATION OF SUPERCRITICAL ACCRETION FLOWS AND A STEADY MODEL WITH OUTFLOWS

We apply our two-dimensional (2D), radially self-similar steady-state accretion flow model to the analysis of hydrodynamic simulation results of supercritical accretion flows. Self-similarity is checked and the input parameters for the model calculation, such as advective factor and heat capacity ratio, are obtained from time-averaged simulation data. Solutions of the model are then calculated and compared with the simulation results. We find that in the converged region of the simulation, excluding the part too close to the black hole, the radial distribution of azimuthal velocity $v_\phi$, density $\rho$ and pressure $p$ basically follows the self-similar assumptions, i.e. they are roughly proportional to $r^{-0.5}$, $r^{-n}$, and $r^{-(n+1)}$, respectively, where $n\sim0.85$ for the mass injection rate of $1000L_\mathrm{E}/c^2$, and $n\sim0.74$ for $3000L_\mathrm{E}/c^2$. The distribution of $v_r$ and $v_\theta$ agrees less with self-similarity, possibly due to convective motions in the $r\theta$ plane. The distribution of velocity, density and pressure in $\theta$ direction obtained by the steady model agrees well with the simulation results {within the calculation boundary of the steady model}. Outward mass flux in the simulations is overall directed toward polar angle of 0.8382 rad ($\sim 48.0^\circ$) for $1000L_\mathrm{E}/c^2$, and 0.7852 rad ($\sim 43.4^\circ$) for $3000L_\mathrm{E}/c^2$, and $\sim$94\% of the mass inflow are driven away as outflow, while outward momentum and energy fluxes are focused around the polar axis. Part of these fluxes lie in the region that are not calculated by the steady model, and special attention should be paid when the model is applied.

Self-Similar Unsteady Flow of a Sisko Fluid in a Cylindrical Tube Undergoing Translation

The governing nonlinear equation for unidirectional flow of a Sisko fluid in a cylindrical tube due to translation of the tube wall is modelled in cylindrical polar coordinates. The exact steady-state solution for the nonlinear problem is obtained. The reduction of the nonlinear initial value problem is carried out by using a similarity transformation. The partial differential equation is transformed into an ordinary differential equation, which is integrated numerically taking into account the influence of the exponentnand the material parameterbof the Sisko fluid. The initial approximation for the fluid velocity on the axis of the cylinder is obtained by matching inner and outer expansions for the fluid velocity. A comparison of the velocity, vorticity, and shear stress of Newtonian and Sisko fluids is presented.

Experimental and theoretical study of dewetting corner flow

AbstractWe study a partial dewetting corner flow with a moving contact line at a finite Reynolds number, $0<\mathit{Re}<O(100)$. When the speed of the moving contact line increases, the receding contact line appears with a corner shape that is also observed in a gravity-driven liquid droplet on an incline and on a plate withdrawn from a bath. In the current problem, $\mathit{Re}\,{\it\epsilon}$ is larger than unity, where ${\it\epsilon}$ is the aspect ratio of the flow structure. Therefore, classical lubrication theory is no longer appropriate. We develop a modified three-dimensional lubrication model for the dewetting corner structure at $\mathit{Re}\,{\it\epsilon}>1$ by taking into account the internal flow pattern and by scaling arguments. The key requirement is that the streamlines in the corner are straight and (nearly) parallel. In this case, we can obtain a modified pressure consisting of the capillary pressure and the dynamic pressure. This model describes the three-dimensional dewetting corner structure at the rear of the moving droplets at $\mathit{Re}\,{\it\epsilon}>1$ and furthermore shows that the dynamic pressure effects become dominant at a small half-opening angle. Additionally, this model provides analytical results for the internal flow, which is a self-similar flow pattern. To validate the analytical results, we perform high-speed shadowgraphy and tomographic particle image velocimetry (PIV). We find a good agreement between the theoretical and the experimental results.

Organised large structure in the post-transition mixing layer. Part 2. Large-eddy simulation

AbstractThree-dimensional large-eddy simulations of two-stream mixing layers developing spatially from laminar boundary layers are presented, replicating wind-tunnel experiments carried out in Part 1 of this study. These simulations have been continued through the mixing transition and into the fully turbulent self-similar flow beyond. In agreement with the experiments, the simulations show that the familiar mechanism of growth by vortex amalgamation is replaced at the mixing transition by a previously unrecognised mechanism in which the spanwise-coherent large structures individually undergo continuous linear growth. In the post-transition flow it is this continuous linear growth of the individual structures that produces the self-similar growth of the mixing-layer thickness, the large-structure interactions occurring as a consequence of the growth, not its cause. New information is also presented on the topography of the organised post-transition flow and on its cyclical evolution through the lifetimes of the individual large structures. The dynamic and kinematic implications of these findings are discussed and shown to define quantitatively the growth rate of the homogeneous post-transition mixing layer in its organised state.

A simple estimator of the Hurst exponent for self-similar traffic flows

In this paper it presents, develops and discusses a new and simple algorithm based on the local maximum likelihood estimator of Whittle, for the estimation of the Hurst exponent of stationary second order time series representing the self-similar with long-range dependence traffic flows presents in the high-speed computer networks. This algorithm reduces the computational cost associated with the calculation of the local maximum likelihood estimator of Whittle reducing the amount of points to evaluate advantage the convexity of the objective function in the interest interval (0.5, 1), and applying a bisection search method over the derivate of the function. The instance is exposed with the intention of being considered as a new and alternative approach for modeling and simulating traffic in existing computer networks.

Dynamics of the envelope of a rapidly rotating star or giant planet in gravitational contraction

We wish to understand the processes that control the fluid flows of a gravitationally contracting and rotating star or giant planet. We consider a spherical shell containing an incompressible fluid that is slowly absorbed by the core so as to mimick gravitational contraction. We also consider the effects of a stable stratification that may also modify the dynamics of a pre-main sequence star of intermediate mass. This simple model reveals the importance of both the Stewartson layer attached to the core and the boundary conditions met by the fluid at the surface of the object. In the case of a pre-main sequence star of intermediate mass where the envelope is stably stratified, shortly after the birth line, the spin-up flow driven by contraction overwhelms the baroclinic flow that would take place otherwise.This model also shows that for a contracting envelope, a self-similar flow of growing amplitude controls the dynamics. It suggests that initial conditions on the birth line are most probably forgotten. Finally, the model shows that the near (Stewartson) layer that lies on the tangent cylinder of the core is likely a key feature of the dynamics that is missing in 1D models.This layer can explain the core and envelope rotational coupling that is required to explain the slow rotation of cores in giant and subgiants stars.

Self-similar turbulent boundary layer with imposed pressure gradient. Four flow regimes

Self-similar flows of an incompressible fluid in a turbulent boundary layer, when the free-stream velocity is a power function (with the exponent m) of the longitudinal coordinate, have been studied. It has been shown that there are four different self-similar flow regimes corresponding to four individual similarity parameters one of which is the known Clauser parameter and the three other parameters have been established for the first time. At adverse pressure gradient, when the exponent m lies in a certain range depending on Reynolds number, the problem has two solutions with different values of the boundary-layer thickness and skin friction; consequently, hysteresis in a pre-separation flow is possible. Separation occurs not at the minimal value of m that corresponds to the strongest adverse pressure gradient, but at m = −0.216 −0.4 Rep−1/3 + O(Rep−2/3), where Rep is the Reynolds number based on longitudinal pressure gradient. The theoretical results are in good agreement with experimental data.

Self-similar flow due to the stretching of a deformable fiber

The boundary layer flow over a stretching fiber in a stationary fluid is investigated by including the coupling of the flow and fiber dynamics. This approach is quite different from the literature on flows over a stretching fiber, which has prescribed fiber kinematics. The flow problem is formulated by using a curvilinear coordinate system, and self-similarity is invoked. The similarity transformations convert the governing partial differential equations of the flow into ordinary differential equations, involving parameters determined by considering the fiber dynamics. The similarity equations are solved numerically using a shooting method. Two kinds of fibers are considered including an elastic one and a viscous one. It is found that the fluid velocity over both kinds decays algebraically to the ambient, though with different rates. Self-similarity imposes restrictions on the parameters of the problem. Nonetheless, practically feasible parameters can be chosen and solutions thereof are presented. The solution can offer more accurate estimations of fiber dynamics and fluid flow for practical applications in fiber stretching.

Enhancement of heat transfer in unsteady laminar oil flow past a heated cylinder at Re = 150

Unsteady convective heat transfer in air and oil flows past a heated circular cylinder is modeled numerically by solving the unsteady Navier—Stokes and energy equations with the aid of multiblock computational technologies implemented in the VP2/3 code using composite overlapping structured grids of different topology. Enhancement of heat and momentum transfer is associated with a considerable reduction of temperature boundary layer thickness. Main attention is paid to a self-similar flow regime and heat transfer, with the analysis of averaged and fluctuation characteristics and the comparison of constant physical property and inhomogeneous media.

Self-similar solution for oblique impact of a water column with sharp front on a wall and its zero inner angle steady limit

The hydrodynamic problem of a three-dimensional (3D) water column impacting on a solid wall is investigated. The focus is on cases of self-similar flow and its related implications in physics. It is demonstrated that limiting process of the self-similar flow can become the steady flow. The problem involves a free surface whose shape is unknown and on which the boundary conditions are nonlinear. It is solved through boundary element method (BEM) with quadrilateral elements within an iterative procedure. The application of the BEM to such a problem has some major challenges. During the process, mesh used in the BEM is regularly regenerated to follow the deformation of the free surface and the data in the old mesh are transferred to the new one. Coordinate rotation technique is used to resolve the difficulties caused by the multi-valued free surface, together with the technique for the thin liquid film over the wall surface. Results are provided for the free surface shapes and pressure distributions for perpendicular and oblique impact cases, including both self-similar flow and its limiting steady flow. Their physics about jet root, the peak pressure and its location, as well as the limiting process to the steady flow are discussed.

On the modulus of continuity for spectral measures in substitution dynamics

The paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms, which yield information about the “fractal” structure of these measures. The main results are, first, a Hölder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hölder estimate for self-similar suspension flows; and, third, a Hölder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hölder estimates for the spectral measures of translation flows along stable foliations of pseudo-Anosov automorphisms. A key technical tool in the proof of the second result is an “arithmetic-Diophantine” proposition, which has other applications. In Appendix A this proposition is used to derive new decay estimates for the Fourier transforms of Bernoulli convolutions.