Steady Solutions Research Articles

Comparison and modification of turbulence models for active flow separation control over a flat surface

The present work studied various models for predicting turbulence in the problem of injecting a fluid microjet into the boundary layer of a turbulent flow. For this purpose, the one-equation Spalart–Allmaras (SA), two-equation k–ε and k–ω, multi-equation transition k-kL–ω, transition shear stress transport (SST), and Reynolds stress models were used for solving the steady microjet into the turbulent boundary layer, and their results are compared with experimental results. Comparing the results indicated that the steady solution methods performed sufficiently we for this problem. Furthermore, it was found that the four-equation transition SST model was the most accurate method for predicting turbulence in this problem. This model predicted the velocity along the x-axis in near- and far-jet locations with about 1% and 5% average errors, respectively. It also outperformed the other methods in predicting Reynolds stresses, especially at the center (nearly 5% error). Moreover, the modified four-equation transition SST model has improved the system's performance in predicting the studied parameters by utilizing Sørensen correlations in predicting Reθt (the transition momentum thickness Reynolds number), Flength (an empirical correlation that controls the length of the transition region), and Reθc (the critical Reynolds number where the intermittency first starts to increase in the boundary layer).

P-Bifurcation Analysis for a Fractional Damping Stochastic Nonlinear Equation with Gaussian White Noise

This work aims to address the P-bifurcation of a stochastic nonlinear system with fractional damping driven by Gaussian white noise. Based on a stochastic averaging method, a fractional damping stochastic nonlinear equation has been studied. Furthermore, the expressions of drift and diffusion coefficients of the Fokker–Planck equation (FPKE) have been obtained. The probability density function (PDF), the steady solution of FPKE, has also been derived. Then, PDFs of two fractional damping Morse oscillators have been obtained. One can note that the analytical results coincide with the results of numerical simulation. Importantly, stochastic P-bifurcation of a fractional damping stochastic nonlinear Morse oscillator has been further addressed and analyzed.

Exponential mixing of Vlasov equations under the effect of gravity and boundary

In this paper, we study exponentially fast mixing induced/enhanced by gravity and stochastic boundary in the kinetic theory of Vlasov equations. We consider the Vlasov equations with and without a vertical magnetic field inside a horizontally-periodic 3D half-space equipped with a non-isothermal diffusive reflection boundary condition of bounded continuous boundary temperature at the bottom. We construct both stationary solutions and global-in-time dynamical solutions in L∞. We prove that moments of a dynamical fluctuation around the steady solutions decay exponentially fast in L∞. As a key of this proof, we establish a uniform bound of so-called residual measures independently of the bouncing number of stochastic characteristics, by constructing a continuous stationary outgoing boundary flux which is strictly positive almost everywhere.

Two-dimensional dynamics of a mobile elliptical cylinder in an upward flow

The fully resolved dynamics of an elliptical particle suspended in an upward flow in an expanding channel, with successively increasing degrees of freedom of motion from a completely fixed state to an eventually fluidized state, are examined using particle resolved direct numerical simulation. The signed distance function immersed boundary method (sdfibm), implemented in OpenFOAM, is validated against results from COMSOL for the case of a pinned ellipse. The aspect ratio of the ellipse (defined as the ratio of the major to the minor axis (γ=Ra/Rb) is held constant at 2, while the minor axis is kept as Rb=0.15 (with respect to the inlet width, Wi = 1). A particle Reynolds number defined as Rep=(2Ra)Vmax/ν (where Vmax = 1 is the centerline velocity) is varied up to 300. The simulations exhibit rich dynamical behavior with stable, steady solutions up to Rep around 67, above which vortex shedding begins, with the ellipse responding with its oscillatory motion in response to the stresses and torques acting on it by the fluid. For the case of free translation in the x direction, multiple oscillatory states are found, where the particle is confined in the left or right half of the flow domain, depending on the initial placement of the particle. The forces on the particle get progressively complicated as the degree of freedom of movement of the ellipse increases, indicating that traditional drag correlations with fixed particles may not be valid in situations when they have mobility if one desires higher fidelity in coarse-grained models.

Gaussian fluctuations around limit measures of generalized SQG point vortices

AbstractWe report on recent results on generalized surface quasi‐geostropic point vortex models. The statistical physics of these models is particularly interesting, as their mean‐field limit is a steady solution of the gSQG PDE. We present a central limit theorem‐type result for gSQG point votex models and make a comparison with results for Euler point vortex models.

Pattern dynamics of a predator–prey system with cross-diffusion, Allee effect and generalized Holling IV functional response

In this paper, a diffusive predator–prey model with Allee effect and functional response of generalized Holling type IV is established. The complex pattern dynamics of and bifurcation phenomena of system are analyzed by using linear stability analyses and bifurcation theory, etc. Corresponding to spiral pattern, spot pattern or spot–stripe mix pattern and chaotic pattern, respectively, the conditions that arise Hopf instability, Turing instability and Hopf–Turing instability of constant positive steady solutions of reaction–diffusion system are presented. The impacts of Allee effect and cross-diffusion on pattern formations of the system are further discussed. Particularly, provided that cross-diffusion is absent, there will not appear the formation of Turing pattern, that is, Turing pattern is only driven by cross-diffusions. Addressed to the different pattern formations involving in cross-diffusion and Allee effect, overall numerical simulations are provided when the spatial dimension is two and the theoretical results are demonstrated. It is revealed that the larger Allee effect constant A and the cross-diffusion coefficient d4 are advantageous for the appearing of formation of Turing pattern, while the cross-diffusion coefficient d3 suppresses the formation of Turing pattern with its increasing.

Existence of Steady Solutions for a Model for Micropolar Electrorheological Fluid Flows with Not Globally log-Hölder Continuous Shear Exponent

In this paper, we study the existence of weak solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted variable exponent Sobolev spaces. We establish the existence of solutions for a material function {hat{p}} that is log -Hölder continuous and an electric field textbf{E} for that vert textbf{E}vert ^2 is bounded and smooth. Note that these conditions do not imply that the variable shear exponent p={hat{p}}circ vert textbf{E}vert ^2 is globally log -Hölder continuous.

Delays and Exposed Populations in Infection Models

Most biological processes take time to occur. In infectious diseases, such as malaria or chikungunya, there is a period of time between when a susceptible individual gets bitten by the vector, and when the individual develops the disease. These times are commonly modeled by introducing delays or by adding exposed as a new population class. Given a model based on differential equations, delays can be introduced in different forms. In this paper, we study different ways of introducing the delays and, alternatively, using exposed populations. We also analyze their steady solutions and stability, and establish the conditions under which the studied models predict an epidemic. Results and conclusions are presented.

Existence of a steady flow through a rotating radial turbine with an arbitrarily large inflow and an artificial boundary condition on the outflow

AbstractWe prove the existence of a steady strong solution to the Navier–Stokes‐type problem in a 2D multiply connected domain, modeling a flow of an incompressible viscous fluid through a rotating radial turbine. We consider the inhomogeneous Dirichlet boundary condition on the inflow Γin and an artificial boundary condition of the “do nothing” type on the outflow Γout. The conditions admit an arbitrarily large flux through the turbine. This is, however, compensated by the requirement that the distance between Γout and the profile family is “sufficiently small”. (Theorem 1.1.) The solution is steady in the rotating frame, that is, in the frame, attached to the rotating turbine. As an auxiliary result, we prove that a function, that takes just two different values in a 1D interval on two complementary measurable sets M and of positive 1D measure, is not in . (Lemma 3.4.)

Symmetry Results for Compactly Supported Steady Solutions of the 2D Euler Equations

In this paper we present symmetry results regarding compactly supported solutions of the 2D steady Euler equations. Assuming that Omega = {x in {mathbb {R}}^2: u(x) ne 0} is an annular domain, we prove that the streamlines of the flow are circular. We are also able to remove the topological condition on Omega if we impose regularity and nondegeneracy assumptions on u at partial Omega . The proof uses the corresponding stream function solves an elliptic semilinear problem -Delta phi = f(phi ) with nabla phi =0 at the boundary. One of the main difficulties in our study is that f is not Lipschitz continuous near the boundary values. However, f(phi ) vanishes at the boundary values and then we can apply a local symmetry result of F. Brock to conclude. In the case partial _{nu } u ne 0 at partial Omega this argument is not possible. In this case we are able to use the moving plane scheme to show symmetry, despite the possible lack of regularity of f. We think that such result is interesting in its own right and will be stated and proved also for higher dimensions. The proof requires the study of maximum principles, The Hopf lemma and The Serrin corner lemma for elliptic linear operators with singular coefficients.

Novel Approach to Improve Stability and Convergence of Flowfield Solution Processes: Mode Multigrid

In the field of computational fluid dynamics, stability and convergence problems are often encountered when solving the governing equation. This paper studies the effect of the mode multigrid on the stability and convergence of iterative algorithms. By further analyzing the mechanism for accelerating the convergence of mode multigrid, a new adaptive mode multigrid (AMMG) is proposed, and an adaptive selection criterion is formulated for the parameters of dynamic modal decomposition, which can accurately identify and filter out the unstable modes in the flowfield, thus efficiently obtaining the accurate steady solution. In the cases of laminar flow past circular cylinder and turbulent flow past airfoil, the AMMG significantly improves the stability of the iterative algorithm; in the case of transonic flow past airfoil, the AMMG solves the problem of shock wave shaking during the iteration, and the convergence is significantly improved. In summary, the AMMG method significantly enhances the stability and convergence of the iterative algorithm, and can be used as an effective convergence-preserving computation strategy.

Penetrative convection: heat transport with marginal stability assumption

This paper investigates heat transport in penetrative convection with a marginally stable temporal-horizontal-averaged field or background field. Assuming that the background field is steady and is stabilised by the nonlinear perturbation terms, we obtain an eigenvalue problem with an unknown background temperature $\tau$ by truncating the nonlinear terms. Using a piecewise profile for $\tau$ , we derived an analytical scaling law for heat transport in penetrative convection as $Ra\rightarrow \infty$ : $Nu=(1/8)(1-T_M)^{5/3}Ra^{1/3}$ ( $Nu$ is the Nusselt number; $Ra$ is the Rayleigh number and $T_M$ corresponds to the temperature at which the density is maximal). A conditional lower bound on $Nu$ , under the marginal stability assumption, is then derived from a variational problem. All the solutions to the full system should deliver a higher heat flux than the lower bound if they satisfy the marginal stability assumption. However, data from the present direct numerical simulations and previous optimal steady solutions by Ding & Wu (J. Fluid Mech., vol. 920, 2021, A48) exhibit smaller $Nu$ than the lower bound at large $Ra$ , indicating that these averaged fields are over-stabilised by the nonlinear terms. To incorporate a more physically plausible constraint to bound heat transport, an alternative approach, i.e. the quasilinear approach is invoked which delivers the highest heat transport and agrees well with Veronis's assumption, i.e. $Nu\sim Ra^{1/3}$ (Astrophys. J., vol. 137, 1963, p. 641). Interestingly, the background temperature $\tau$ yielded by the quasilinear approach can be non-unique when instability is subcritical.

A Method for Predicting the Aeroelastic Response of Radial Turbomachines

AbstractAircraft intermittent combustion engines often incorporate turbochargers adapted from ground-based applications to improve their efficiency and performance. These turbochargers can operate at off-design conditions and experience blade failures brought on by aerodynamic-induced blade resonances. A reduced-order model of the aeroelastic response of general fluid–structural configurations is developed using the Euler–Lagrange equation informed by numerical data from uncoupled computational fluid dynamic (CFD) and computational structural dynamic calculations. The structural response is derived from a method of assumed-modes approach. The unsteady fluid response is described by a modified version of piston theory that approximates the local transient pressure fluctuation in conjunction with steady CFD solution data. The reduced-order model is first applied to a classical panel flutter scenario and found to predict a flutter boundary that compares favorably to the boundary identified by existing theory and experimental data. The model is then applied to the high-pressure turbine of a dual-stage turbocharger. The model predictions are shown to reliably determine the lack of turbine blade flutter and rudimentary damping comparisons are performed to assess the ability of the model to ascertain the susceptibility of the turbine to forced response. Obstacles associated with the current experimental state of the art that impinge upon further numerical validation are discussed.

Investigation of non-ideal gas flows around a circular cylinder

The aerodynamic performance of a cylinder Pitot probe for velocity measurements in compressible non-ideal gas flows, such as those encountered in Organic Rankine Cycle (ORC) turbines, is investigated by means of Computational Fluid Dynamics. Numerical simulations are performed at subsonic and transonic conditions, and freestream Reynolds numbers are in the cylinder critical regime. The working fluid is the organic vapor Novec™ 649. Air flow simulations at similar inlet conditions are reported for comparison. Steady and unsteady RANS solutions are computed with the Spalart–Allmaras turbulence model. The results are assessed against experimental measurements collected in a wind tunnel. URANS is in good agreement with experimental data for all considered conditions, and delivers reasonably accurate estimations of the cylinder back pressure. Using a dense gas leads to a lower minimum pressure coefficient compared to air, alongside a reduced maximum Mach number due to the non-ideal speed of sound behaviour. In the experimentally studied range of compressibility factors and Mach numbers, discrepancies observed with respect to air flow are mostly an effect of the different isentropic exponents. In the transonic regime, shock waves causing boundary layer separation are weakened in the dense gas, but back pressure is also decreased, contributing to rising form drag.

Stability of the 3D Boussinesq equations with partial dissipation near the hydrostatic balance

The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations but also are physically relevant and mathematically important. Unfortunately, it is not often well‐understood for many ranges of fractional powers. This paper focuses on a system of the 3D Boussinesq equations with fractional horizontal and dissipation and proves that if the initial data ( ) in the Sobolev space are close enough to the hydrostatic balance state, respectively, the equations with then always lead to a steady solution.

Global Instability of Rod Eutectic Growth in Directional Solidification

In our previous work, we obtained the uniformly valid asymptotic solution of a cylindrical rod eutectic. In order to further study the critical point of the stable growth of a rod eutectic, we have considered the unsteady growth of a rod eutectic on the basis of the steady solution of the rod eutectic. Based on the experimental system of rod eutectic growth, combined with solidification thermodynamics and kinetics, the unsteady mathematical model of the rod eutectic was established. We used the asymptotic analysis method to seek the analytical solution of the mathematical model and used the nonlinear stability analysis theory to analyze the analytical solution and establish the corresponding disturbance model. We obtained the analytic form of the global mode solution and the corresponding quantization conditions and find that there is a stable growth mode, namely the mode (ST-mode), for rod eutectic growth; when ε<εST0, the rod eutectic growth is stable, when ε>εST0, the rod eutectic growth is unstable and when ε=εST0, the rod eutectic growth is of a neutral stability. The critical eutectic spacing of succinonitrile(D)camphor (SCN-DC) predicted by us is smaller than that predicted by Jackson–Hunt, which is consistent with the corresponding experimental data. Finally, we found that the critical eutectic spacing and stable region of rod eutectic growth changed little with the temperature gradient.

Krylov Methods for Large-Scale Dynamical Systems: Application in Fluid Dynamics

AbstractIn fluid dynamics, predicting and characterizing bifurcations, from the onset of unsteadiness to the transition to turbulence, is of critical importance for both academic and industrial applications. Different tools from dynamical systems theory can be used for this purpose. In this review, we present a concise theoretical and numerical framework focusing on practical aspects of the computation and stability analyses of steady and time-periodic solutions, with emphasis on high-dimensional systems such as those arising from the spatial discretization of the Navier–Stokes equations. Using a matrix-free approach based on Krylov methods, we extend the capabilities of the open-source high-performance spectral element-based time-stepper Nek5000. The numerical methods discussed are implemented in nekStab, an open-source and user-friendly add-on toolbox dedicated to the study of stability properties of flows in complex three-dimensional geometries. The performance and accuracy of the methods are illustrated and examined using standard benchmarks from the fluid mechanics literature. Thanks to its flexibility and domain-agnostic nature, the methodology presented in this work can be applied to develop similar toolboxes for other solvers, most importantly outside the field of fluid mechanics.

Forward-modulated damping estimates and nonlocalized stability of periodic Lugiato–Lefever waves

In an interesting recent analysis, Haragus–Johnson–Perkins–de Rijk have shown modulational stability under localized perturbations of steady periodic solutions of the Lugiato–Lefever equation (LLE), in the process pointing out a difficulty in obtaining standard “nonlinear damping estimates” on modulated perturbation variables to control regularity of solutions. Here, we point out that in place of standard “inverse-modulated” damping estimates, one can alternatively carry out a damping estimate on the “forward-modulated” perturbation, noting that norms of forward- and inverse-modulated variables are equivalent modulo absorbable errors, thus recovering the classical argument structure of Johnson–Noble–Rodrigues–Zumbrun for parabolic systems. This observation seems of general use in situations of delicate regularity. Applied in the context of (LLE), it gives the stronger result of stability and asymptotic behavior with respect to nonlocalized perturbations.

Global stability analysis of bubbles rising in a vertical capillary with an external flow

We study the linear stability of bubbles in a capillary tube under external flow. Yu et al. (J. Fluid Mech., vol. 911, 2021, pp. 1–19) showed that a rich variety of bubble dynamics occurs when a downward external flow is applied, opposing the buoyancy-driven ascent of the bubble. They found experimentally and numerically the existence of two branches of solutions that overlap over a finite range of the capillary number of the downward external flow in cases where the Reynolds number is small and the Bond number is larger than the critical value for which the bubble can rise spontaneously (Bretherton, J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166–188). Furthermore, inertialess, symmetry-breaking steady-state shapes were found as the bubble transits near the tipping points of the solution branches. In this work, using steady axisymmetric simulations, we show that the reported multiplicity of solutions can be described using bifurcation diagrams with three branches of steady axisymmetric solutions and two limit points. The linear global stability analysis of the different branches of the stationary axisymmetric solutions demonstrates that the symmetry breaking is due to the development of three-dimensional instabilities with azimuthal wavenumber $|m|=1$ .

Non-destructive evaluation and fast conductivity calculation of various nanowire-based thin films with artificial neural network aided THz time-domain spectroscopy.

Terahertz time-domain spectroscopy (THz-TDS) has been utilized extensively to characterize materials in a non-destructive way. However, when materials are characterized with THz-TDS, there are many extensive steps for analyzing the acquired terahertz signals to extract the material information. In this work, we present a significantly effective, steady, and rapid solution to obtain the conductivity of nanowire-based conducting thin films by utilizing the power of artificial intelligence (AI) techniques with THz-TDS to minimize the analyzing steps by training neural networks with time domain waveform as an input data instead of a frequency domain spectrum. For this purpose, Al-doped and undoped ZnO nanowires (NWs) on sapphire substrates and silver nanowires (AgNWs) on polyethylene terephthalate (PET) and polyimide (PI) substrates have been measured for dataset creation via THz-TDS. After training and testing a shallow neural network (SSN) and a deep neural network (DNN) to obtain the optimum model, we calculated conductivity in a conventional way, and the prediction based on our models matched successfully. This study revealed that users could determine a sample's conductivity without fast Fourier transform and conventional conductivity calculation steps within seconds after obtaining its THz-TDS waveform, demonstrating that AI techniques have great potential in terahertz technology.