Minimum Critical Reynolds Number Research Articles

Stability of plane Couette flow with constant wall transpiration

Plane Couette flow with constant wall transpiration, i.e., constant blowing from below and constant suction from above, is a solution of the Navier-Stokes equations in terms of exponentials. It is characterized by two Reynolds numbers Re and ReV, where Re parametrizes the moving wall and ReV describes the influence of the transpiration. For this flow the modified Orr-Sommerfeld equation admits one of the very few exact solutions in terms of hypergeometric functions. Based on this, we solve the stability problem, though for the main part focus on temporal stability. For small wall-transpiration rates up to ReV≃6.71, the flow remains unconditionally stable, analogous to the classical Couette flow for arbitrary Reynolds numbers Re. By further increasing ReV an instability sets in and a minimum critical Reynolds number of Re=668350.491 is reached at ReV=9.799. Hence, around this point, the destabilizing effect of blowing outweighs the stabilizing effect of suction. By further increasing the transpiration rate beyond this point, the corresponding critical Reynolds number Re increases again and continues to grow. This limiting case is accompanied by the development of a strong boundary-layer-like velocity profile near the upper wall and the flow transitions to the asymptotic suction boundary layer (ASBL). Thus, the present analysis comprises the whole range from classical Couette flows extended by transpiration to the ASBL, which is known to have a strongly stabilizing effect. Published by the American Physical Society 2024

The effects of particle shape, orientation, and Reynolds number on particle-wall collisions

Crystallisation is an important unit operation used in the production of Active Pharmaceutical Ingredients (APIs). Crystallisation is typically carried out using seed crystals in a process called secondary nucleation that allows for greater control of crystal quality attributes. The mechanism of secondary nucleation is still not well understood. Particle attrition is one proposed mechanism. This work examines the conditions under which particle-wall collisions occur. This is done through the simulation of free-moving particles in an impinging jet flow using an immersed-boundary lattice Boltzmann method (IB-LBM) CFD solver. Particle Reynolds numbers from 100 to 400 are examined. Particle shapes with aspect ratios from 1:1 to 8:1 are used to represent the crystal habits of APIs.Particles that start in their low-drag or high-drag orientation maintain this orientation on approach to the target surface. All other intermediate initial orientations examined cause particles to rotate toward and overshoot their high-drag form before adopting their high-drag form in proximity to the target surface. Particles in their low-drag form remain in their initial orientation due to the symmetry of the computational domain. In this orientation, a collision is most likely to occur, and the minimum critical Reynolds number at which a particle-wall collision will occur can be determined. This value is shown to increase with increasing particle frontal length. Pointed or rounded leading edges are shown to improve a particle's ability to pierce the boundary layer adjacent to the target surface and reduce this value. In cases where a Reynolds number of 400 is insufficient to cause a particle-wall collision, the particles’ minimum distances from the target surface are reported. Using the minimum critical Reynolds number values obtained, the approach velocities required for particles to collide with a wall are shown to be larger than the impeller tip speeds typically used during crystallisation operations. This work provides for the first time the conditions under which particle-wall collisions occur for varying shape and orientation, their behaviour on approach, and the associated impact velocities.

Experimental study on effect of structural parameters on pressure drop characteristics of multi-orifice plates

Owing to the advantages of rectification, low noise, low pressure loss, and more stable pressure drop signal compared with the characteristics of the conventional single-orifice plate, multi-orifice plates (MOPs) offer a wide range of application prospects in various industrial pipelines. However, owing to its various orifice shapes and orifice layouts, the pressure drop characteristics of MOPs are not fully understood. In this study, the pressure drop characteristics of single-phase flow across a multi-orifice plate, which implies a stable zone (turbulent flow) pressure loss coefficient and the minimum critical Reynolds number of the stable zone, are experimentally investigated in the Reynolds number range of 29,000–146,000 using water as the fluid. Nine MOPs with circular holes evenly arranged in an equilateral triangle are tested. Their structures differ in terms of the equivalent diameter ratio (0.30–0.60), orifice number (64–400), and relative orifice thickness (0.40–1.60). The results show that the minimum critical Reynolds number deceases as the equivalent diameter ratio decreases, the relative orifice thickness increases, and the orifice number increases, which allows an enlarged Reynolds number range for the stable zone. As the equivalent diameter ratio, orifice number, and relative orifice thickness increase, the stable zone pressure loss coefficient of MOPs initially decreases rapidly and gradually approaches a constant. Finally, based on the experimental results of this study, a correlation of the stable zone pressure loss coefficient of MOPs is proposed, which provides a reference for using MOPs in various industrial applications.

Quantification of fabric morphology and prediction of drag crisis based on the fractal dimension

This paper focused on investigating the relationship between fabric morphology and aerodynamic drag. The fractal theory is used to quantify the fabric surface morphology, and its influence on the minimum drag coefficient and critical Reynolds number is evaluated. In addition, the effectiveness of the fractal dimension in predicting the drag crisis is verified. It is observed that with the increase of the fractal dimension, the minimum drag coefficient increases, and the correlation coefficient can reach 0.9728, while the corresponding critical Reynolds number decreases, and the correlation coefficient can reach 0.9334. In addition, the accuracy of predicting drag crisis by the fractal dimension is verified and the relative error of predicting the drag crisis by the fractal dimension is less than 10%, which is significantly better than that by roughness.

Fundamental scales in the kinematic phase of the turbulent dynamo

ABSTRACT The turbulent dynamo is a powerful mechanism that converts turbulent kinetic energy to magnetic energy. A key question regarding the magnetic field amplification by turbulence, is, on what scale, kp, do magnetic fields become most concentrated? There has been some disagreement about whether kp is controlled by the viscous scale, kν (where turbulent kinetic energy dissipates), or the resistive scale, kη (where magnetic fields dissipate). Here, we use direct numerical simulations of magnetohydrodynamic turbulence to measure characteristic scales in the kinematic phase of the turbulent dynamo. We run 104-simulations with hydrodynamic Reynolds numbers of 10 ≤ Re ≤ 3600, and magnetic Reynolds numbers of 270 ≤ Rm ≤ 4000, to explore the dependence of kp on kν and kη. Using physically motivated models for the kinetic and magnetic energy spectra, we measure kν, kη, and kp, making sure that the obtained scales are numerically converged. We determine the overall dissipation scale relations $k_\nu = (0.025^{+0.005}_{-0.006})\, k_\text{turb}\, \mbox{Re}^{3/4}$ and $k_\eta = (0.88^{+0.21}_{-0.23})\, k_\nu \, \mbox{Pm}^{1/2}$, where kturb is the turbulence driving wavenumber and Pm = Rm/Re is the magnetic Prandtl number. We demonstrate that the principle dependence of kp is on kη. For plasmas, where Re ≳ 100, we find that $k_p= (1.2_{-0.2}^{+0.2})\, k_\eta$, with the proportionality constant related to the power-law ‘Kazantsev’ exponent of the magnetic power spectrum. Throughout this study, we find a dichotomy in the fundamental properties of the dynamo where Re > 100, compared to Re < 100. We report a minimum critical hydrodynamic Reynolds number, Recrit = 100 for bonafide turbulent dynamo action.

Flow instability and heat transfer enhancement of unsteady convection in a step channel

The formation of flow instability and heat transfer enhancement of unsteady convection is studied for backward-facing step flow, forward-facing step flow and combined step flow. The governing equations are solved by a program of FORTRAN code. The numerical method is validated by PIV and heat transfer experiments. The effects and the affecting factors of flow instability are investigated. The results show that the flow instability caused by the reattachment flow has a positive effect on heat transfer. In combined step flow, the formation of this flow instability is mainly affected by the bottom wall length and the critical Reynolds number. The minimum critical Reynolds number of the transition from steady to unsteady case is between 200 and 250. The fundamental flow pattern results show that 7 typical flow characters appear in different step channel. A new concept named the vortex length ratio (LRv) is proposed to distinguish the vortex shape. The vortex with the smaller LRv is more effective in heat transfer enhancement mainly because of the increase of vortex attack angle and vortex rotating speed.

The convective instability of the boundary-layer flow over a rotating cone in and out of a uniform magnetic field

We present convective instability analyses of the boundary-layer flows originated over cones (with half-angle ψ≤90∘) rotating in an otherwise still conducting fluid. A uniform magnetic field (with magnetic strength parameter, m≥0) is acting normally on the surface of each cone. In the non-magnetic case of cones, comparison of present results with existing experimental and theoretical studies lead *to propose that onset of instabilities may be attributed to crossflow (type I) modes and streamline curvature (type II) modes for broad cones (with ψ≥40∘) and more slender cones (with ψ≤15∘), respectively. For slender cones with half-angle ψ in the vicinity of 25° the presented results validate the hypothesis of centrifugal (type III) modes ( discovered by Garrett et al. (2014)). In the magnetic case, increasing magnetic strength m∈(0,11] considerably delays the onset of instabilities (due to both type I and type II modes) over each rotating cone with fixed ψ. The stabilising influence of magnetism is in agreement with the meagrely available theoretical studies for magnetic rotating disk case ψ=90∘. In addition, our results lead to suggest that streamline curvature mode over each cone with fixed half-angle ψ become sensitive with increasing m. A minimum m0∈[0,11] exists for each fixed half-angle ψ0≤90∘ such that whenever m≥m0 the onset of instabilities (in the sense of minimum critical Reynolds numbers) over every cone with half-angle (ψ≤ψ0) are commenced by type II mode instead of type I mode whilst the aforementioned conjecture of stabilising effect of magnetism is not violated in the considered range of parameters. Under non-stationary vortices assumption with magnetism, we show that crossflow modes travelling at around 75% of the cone surface are likely to be selected in applications where smooth and frictionless surfaces are used. This finding is in complete agreement with the non-magnetic case of rotating cone in existing studies.

Laminar-turbulent transition in channel flow: wall effects and critical Reynolds number

This article describes a possible cause of natural laminar-turbulent transition in channel flow, and the minimum critical Reynolds number Rc,min is determined. It is assumed that the mechanism of transition is the same for both circular pipe flow and channel flow since each flow has its own minimum critical Reynolds number. Our starting points are that under natural disturbance conditions, transition appears to take place only in the developing entrance region and that the critical Reynolds number Rc becomes Rc,min when using a sharp-edged uniform channel. In our previous studies of circular pipe flow, we have developed a model for transition and obtained Rc,min=1910 and 1950 in two mesh systems. In this study, for channel flow, the above transition model is verified by obtaining Rc,min=1190 and 1260 in two mesh systems.

Use of natural instabilities for generation of streamwise vortices in a laminar channel flow

An analysis of pressure-gradient-driven flows in channels with walls modified by transverse ribs has been carried out. The ribs have been introduced intentionally in order to generate streamwise vortices through centrifugally driven instabilities. The cost of their introduction, i.e. the additional pressure losses, have been determined. Linear stability theory has been used to determine conditions required for the formation of the vortices. It has been demonstrated that there exists a finite range of rib wave numbers capable of creating vortices. Within this range, there exists an optimal wave number which results in the minimum critical Reynolds number for the specified rib amplitude. The optimal wave numbers marginally depend on the rib positions and amplitudes. As the formation of the vortices can be interfered with by viscosity-driven instabilities, the critical conditions for the onset of such instabilities have also been determined. The rib geometries which result in the vortex formation with the smallest drag penalty and without interference from the viscosity-driven instabilities have been identified.

Evolution of length scales and statistics of Richtmyer-Meshkov instability from direct numerical simulations

In this study we present direct numerical simulation results of the Richtmyer-Meshkov instability (RMI) initiated by Ma=1.05,Ma=1.2, and Ma=1.5 shock waves interacting with a perturbed planar interface between air and SF(6). At the lowest shock Mach number the fluids slowly mix due to viscous diffusion, whereas at the highest shock Mach number the mixing zone becomes turbulent. When a minimum critical Taylor microscale Reynolds number is exceeded, an inertial range spectrum emerges, providing further evidence of transition to turbulence. The scales of turbulent motion, i.e., the Kolmogorov length scale, the Taylor microscale, and the integral length, scale are presented. The separation of these scales is found to increase as the Reynolds number is increased. Turbulence statistics, i.e., the probability density functions of the velocity and its longitudinal and transverse derivatives, show a self-similar decay and thus that turbulence evolving from RMI is not fundamentally different from isotropic turbulence, though nominally being only isotropic and homogeneous in the transverse directions.

Critical Reynolds number of the couette flow of a vibrationally excited diatomic gas energy approach

An energy balance equation for plane-parallel flows of a vibrationally excited diatomic gas described by a two-temperature relaxation model is derived within the framework of the nonlinear energy theory of hydrodynamic stability. A variational problem of calculating critical Reynolds numbers Recr determining the lower boundary of the possible beginning of the laminar-turbulent transition is considered for this equation. Asymptotic estimates of Recr are obtained, which show the characteristic dependences of the critical Reynolds number on the Mach number, bulk viscosity, and relaxation time. A problem for arbitrary wave numbers is solved by the collocation method. In the realistic range of flow parameters for a diatomic gas, the minimum critical Reynolds numbers are reached on modes of streamwise disturbances and increase approximately by a factor of 2.5 as the flow parameters increase.

Energy Estimate of the Critical Reynolds Numbers in a Compressible Couette Flow. Effect of Bulk Viscosity

A variational problem of determining the critical Reynolds number of the laminar-turbulent transition is numerically solved within the framework of the nonlinear energy theory of stability of compressible flows. Stability of various modes in the Couette flow of a compressible gas is estimated by the method of collocations. It is demonstrated that the minimum critical Reynolds numbers in the range of the ratio of the bulk viscosity ηb to the shear viscosity η, which is realistic for diatomic gases, are reached for modes of streamwise disturbances. The critical Reynolds numbers increase as the bulk viscosity is increased in the interval ηb = 0-2η, with the maximum increase in the limit being approximately 30%.

Traveling wave instability in a diverging–converging channel

The temporal linear stability of a pressure-driven flow in a diverging–converging channel with respect to the traveling wave disturbances is considered. The flow may develop under either the fixed mass flowrate constraint or the fixed pressure gradient constraint. It is shown that the variations in the channel geometry lead to the flow destabilization, as compared with the straight channel case. It is demonstrated that the critical disturbances have the form of two-dimensional waves. The global critical conditions describing the minimum critical Reynolds number required to create the instability for the specified geometry of the channel are given. It is shown that the flow developed under the fixed mass flowrate constraint is slightly more unstable than the flow developed under the fixed pressure gradient constraint. This difference increases with an increase of the amplitude of the channel divergence–convergence.

Some interesting consequences of the maximum entropy production principle

Two nonequilibrium phase transitions (morphological and hydrodynamic) are analyzed by applying the maximum entropy production principle. Quantitative analysis is for the first time compared with experiment. Nonequilibrium crystallization of ice and laminar-turbulent flow transition in a circular pipe are examined as examples of morphological and hydrodynamic transitions, respectively. For the latter transition, a minimum critical Reynolds number of 1200 is predicted. A discussion of this important and interesting result is presented.

Hydromagnetic stability of plane Couette flow of an upper convected Maxwell fluid

Hydrodynamic stability of plane Couette flow of an upper convected Maxwell fluid is investigated in presence of a transverse magnetic field assuming that the magnetic Prandtl number is sufficiently small. The resulting equation is a modified Orr-Sommerfeld equation. The equations of stability are solved numerically using Chebyshev collocation method with QZ algorithm. The critical values of Reynolds number, wave number and wave speed are computed and the results are shown through the neutral curves. By increasing the amount of elasticity to a certain value, it is shown that, as the Hartmann number increases, the minimum critical Reynolds number decreases and it does not increase again in contrast to the Newtonian case.

Wall-transpiration-induced instabilities in plane Couette flow

Linear stability of Couette flow modified by transpiration applied at the lower wall is considered. It is shown that transpiration can induce flow instability resulting in the appearance of streamwise-vortex-like structures. It is argued that the instability is driven by centrifugal forces associated with streamline curvature. The conditions leading to the onset of the instability depend on the amplitude and wavelength of the transpiration and can be expressed in terms of the critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number required for the onset of the instability for the specified amplitude of the transpiration regardless of its wavelength are also given. The threshold amplitude required for the onset varies approximately as .

Weakly nonlinear stability of Hartmann boundary layers

By means of a weakly nonlinear stability analysis it is shown that the Hartmann boundary layer presents subcritical instability in the proximity of the minimum linear critical Reynolds number. This gives further support to earlier speculations that finite amplitude effects account for the discrepancies between the results of the linear stability analysis and experimental evidence on laminarisation.

Vortex instability in a divergingconverging channel

Linear stability of flow in a diverging–converging channel is considered. The flow may develop under either the fixed mass or the fixed pressure gradient constraint. Both cases are considered. It is shown that under certain conditions the divergence–convergence of the channel leads to the formation of a secondary flow in the form of streamwise vortices. It is argued that the instability is driven by centrifugal effect. The instability has two modes and conditions leading to their onset have been identified. These conditions depend on the amplitude and the length of the channel diverging–converging section and can be expressed in terms of a critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number required to create the instability for the specified amplitude of the variations of the channel opening are also given. It is shown that the flow developed under the fixed mass constraint is slightly more unstable than the flow developed under the fixed pressure constraint. This difference increases with an increase of the amplitude of the channel divergence–convergence.

平行平板間・助走区間の流れ場の数値解析 （２）最小臨界レイノルズ数

平行平板間・助走区間の流れ場の数値解析 （２）最小臨界レイノルズ数

1547 層流乱流遷移決定モデルの検証 : 平行平板間流

A conceptual model has been constructed for the problem of determining a critical Reynolds number Rc from laminar to turbulent flows in a circular pipe and between parallel plates (Kanda [2-3]). For circular pipe flow, the minimum critical Reynolds number Rc(min) of approximately 2040 was obtained. In order to prove the validity of the model, another verification is required. Thus, for flow between parallel plates, results of previous investigations were studied, focusing on experimental data on Rc, the entrance length, and the transition length. Consequently, the model was confirmed and an experimental value of Rc(min) was found to exist in the neighborhood of 1300,based on the channel height and average velocity. For flow between parallel plates, the author obtained Rc(min) of approximately 1230 when using J0=101 grid points in the y-direction, and 910 when J0=51.