Two-dimensional Flow Research Articles

A study of streamline geometries in subsonic and supersonic regions of compressible flow fields

The geometrical properties of streamlines, such as the curvatures, directions and positions, are studied in steady inviscid compressible flow fields via differential geometry theories and conservation laws. The influences of the streamline geometries on the flow speeds and pressures are also identified and discussed. By transforming the streamlines to fill the domain and satisfy the boundary conditions, a unified geometry-based solver, the streamline transformation method, is proposed for both subsonic and supersonic regions. The governing equations and boundary conditions along streamlines and shock waves are also derived. This method is verified by numerical results of three typical flow fields, including the subsonic channel flow, the supersonic downstream of attached shock waves and especially the subsonic/supersonic downstream of detached bow shock waves. Both two-dimensional planar and axisymmetric flow fields are considered. Compared with the results from computational fluid dynamics, good agreements are achieved by this method, while fewer computational resources, by an order of magnitude, are consumed. Features of these flow fields are also analysed from a geometrical perspective, such as flow speeds and pressures deviated by the wall curvatures, and three-dimensional effects in the after-shock flow fields. For a hyperbolic-shaped bow shock wave, the stand-off distances and the transitions from subsonic to supersonic regions are also discussed. As indicated by the accuracy, efficiency and applicability in a wide range of flow speeds, the streamline transformation method would be a potential candidate for the theoretical analysis and inverse design of high-speed flow fields, especially where the subsonic regions exist downstream of strong shock waves.

Implication of injection angle and jet-to-cross flow ratio on the combustion characteristics of a scramjet combustor

The present work deals with the implication of injection angle and jet-to-cross flow ratio on the combustor performance of a cavity-based scramjet combustor. Unsteady, two-dimensional compressible flow through the scramjet combustor is considered to comprehend the effects of Mach number (Ma), jet-to-cross flow ratio ([Formula: see text]) and injection angle (α) on the combustion characteristics. Results are presented for the following range of parameters: 2.0 ≤ Ma ≤ 2.7; 15° ≤ α ≤ 60° and 0.5 ≤ [Formula: see text] ≤ 1.2. Intense recirculation zone is observed at θ = 45° irrespective of inlet Mach number which are quintessential for proper mixing. Further, the size of the recirculation zone is the largest when jet-to-cross flow ratio is equal to 0.8 acting as a precursor for enhanced mixing performance. Moreover, static pressure reaches an optimum value at injection angle of 45° beyond which further increase would result in thermal choking. Our study reveals that mixing is improved with an increase in Mach number (Ma). It is observed that the size of the recirculation zone is strongly dependent on the jet-to-cross flow pressure ratio along with the injection angle. To put it in a nutshell, the injection angle of 45° along with a jet-to-cross flow pressure ratio of 0.8 is conducive the optimal performance in scramjet combustor.

Study on heat transfer and pressure steady-state characteristics of a floating nozzle under a moving wall

This work considers the flow field as two-dimensional turbulent flow and studies the steady-state properties of heat transfer and the pressure of the suspension nozzle. An adiabatic wall parallel to the moving wall and two slit entrances at either end of the adiabatic wall make up the rectangular flow field. The SST k-ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k - \\omega$$\\end{document} turbulence model is used in the turbulence computation. Both qualitative and quantitative analyses are conducted on the distribution of the flow field, temperature field, local Nusselt number, local pressure coefficient, average Nusselt number, and average pressure coefficient under various combination conditions. The findings indicate that when the suspension nozzle's flow field varies greatly, wall-jet velocity ratio is 0.1. A rise in Jet inclination angle is not helpful for the wall's suspension, and it has minimal effect on the flow field. The flow field is greatly influenced by separation space-slit width ratio. Larger separation space-slit width ratio values are advantageous for the wall's heat transmission but unfavorable for the wall's suspension. The flow field is most influenced by wall-jet velocity ratio. The wall's ability to convey heat is stronger the higher the wall-jet velocity ratio, but its ability to support weight falls.

Two-dimensional Maxwell hybrid nanofluid flow subject to heat source/sink and convective conditions across a stretching surface: an application of parametric continuation method

Two-dimensional Maxwell hybrid nanofluid flow subject to heat source/sink and convective conditions across a stretching surface: an application of parametric continuation method

The viscous force and torque on a closed irrotational surface

The force and torque on a solid body in a viscous potential flow are often taken to be independent of viscosity. Joseph et al. (Eur. J. Mech. B/Fluids, vol. 12, 1993, pp. 97–106; J. Fluid Mech., vol. 265, 1994, pp. 1–23) proved that this holds for (i) the force (not the torque) in any two-dimensional flow, and (ii) the drag force experienced by a purely translating three-dimensional body. The remaining components of the force and torque, along with general three-dimensional flows, were not considered. Importantly, the flow was assumed to be unbounded and irrotational everywhere. We eliminate this rarely satisfied assumption and consider the viscous force and torque experienced by any closed surface where the flow is irrotational locally; this can include a body's surface. Any vorticity distribution is permitted away from the closed irrotational surface. In so doing, we complete the analysis of Joseph et al. for all components of the viscous force and torque in two and three dimensions and enable application to real flows that inevitably contain regions of vorticity.

Wake Structures and Hydrodynamic Characteristics of Flows around Two Near-Wall Cylinders in Tandem and Parallel Arrangements

To clarify the hydrodynamic interference characteristics of flows around multiple cylinders under the wall effect, the two-dimensional (2D) flows around the near-wall single, two tandem and parallel cylinders are simulated under different gap ratios (0.15 ≤ G/D ≤ 3.0) and spacing ratios (1.5 ≤ T/D ≤ 4.0) at a Reynolds number of Re = 6300. We also examine the wake patterns, the force coefficients, and the vortex-shedding frequency with emphases on the wall effect and effects of the two-cylinder interference. A critical wall gap of G/D = 0.6 is identified in the single-cylinder case where the wall can exert significant influences. The two near-wall tandem cylinders exhibit three wake states: stretching mode, attachment mode, and impinging mode. The force coefficients on the upstream cylinder are significantly affected by the wall for G/D ≤ 0.6. The downstream cylinder is mainly influenced by the upstream cylinder. For G/D > 0.6, the force coefficients on the two cylinders exhibit a similar variation trend. In the parallel arrangement, the two cylinders exhibit four wake states in different G/D and T/D ranges: double stretching mode, hetero-vortex scale mode, unilateral vortex mode, and free vortex mode. Moreover, the two parallel cylinders in the hetero-vortex scale or free vortex mode have two states: synchronous in-phase state and synchronous out-of-phase state. The mean drag coefficients on the two cylinders decrease, while the mean lift coefficients exhibit opposite variation trends, as the T/D grows.

Unwrapping phase contrast MRI by iterative graph cuts.

To develop and evaluate a phase unwrapping method for cine phase contrast MRI based on graph cuts. A proposed Iterative Graph Cuts method was evaluated in 10 cardiac patients with two-dimensional flow quantification which was repeated at low venc settings to provoke wrapping. The images were also unwrapped by a path-following method (ROMEO), and a Laplacian-based method (LP). Net flow was quantified using semi-automatic vessel segmentation. High venc images were also wrapped retrospectively to asses the residual amount of wrapped voxels. The absolute net flow error after unwrapping at venc = 100cm/s was 1.8mL, which was 0.83mL smaller than for LP. The repeatability error at high venc without unwrapping was 2.5mL. The error at venc = 50cm/s was 7.5mL, which was 8.2mL smaller than for ROMEO and 5.7mL smaller than for LP. For retrospectively wrapped images with synthetic venc of 100/50/25cm/s, the residual amount of wrapped voxels was 0.00/0.12/0.79%, which was 0.09/0.26/8.0percentage points smaller than for LP. With synthetic venc of 25cm/s, omitting magnitude information resulted in 3.2 percentage points more wrapped voxels, and only spatial/temporal unwrapping resulted in 4.6/21 percentage points more wrapped voxels compared to spatiotemporal unwrapping. Iterative Graph Cuts enables unwrapping of cine phase contrast MRI with very small errors, except for at extreme blood velocities, with equal or better performance compared to ROMEO and LP. The use of magnitude information and spatiotemporal unwrapping is recommended.

Computationally Efficient and Interface Accurate Dual-Grid Phase-Field Simulation of Turbulent Drop-Laden Flows

Abstract In this work, we develop a dual-grid approach for the direct numerical simulations (DNS) of tur- bulent multiphase flows in the framework of the phase-field method (PFM). With the dual-grid approach, the solution of the Navier-Stokes equations (flow-field) and of the Cahn-Hilliard equa- tion (phase-field) are performed on two different computational grids. In particular, a base grid - fine enough to resolve the flow down to the Kolmogorov scale - is used for the solution of the Navier-Stokes equations, while a refined grid - required to improve the description of small interfacial structures - is used for the solution of the Cahn-Hilliard equation (phase-field method). The proposed approach is validated, and its computational efficiency is evaluated considering the deformation of a drop in a two-dimensional shear flow. Analyzing the computational time and memory usage, we observe a reduction between ≃30% and ≃40% (with respect to the single-grid approach), depending on the grid refinement factor employed for the phase-field variable. The applicability of the approach to a realistic three-dimensional case is also discussed, by focusing on the breakage of a thin liquid sheet inside a turbulent channel flow. Indications on the grid resolution representing a good compromise between accuracy and computational efficiency in drop-laden turbulence are also provided.

Implementation of IDEAL algorithm based on Delaunay triangular mesh for 2D-compressible flows

PurposeTo present the detailed implementation processes of the IDEAL algorithm for two-dimensional compressible flows based on Delaunay triangular mesh, and compare the performance of the SIMPLE and IDEAL algorithms for solving compressible problems. What’s more, the implementation processes of Delaunay mesh generation and derivation of the pressure correction equation are also introduced.Design/methodology/approachProgramming completely in C++.FindingsFive compressible examples are used to test the SIMPLE and IDEAL algorithms, and the comparison with measurement data shows good agreement. The IDEAL algorithm has much better performance in both convergence rate and stability over the SIMPLE algorithm.Originality/valueThe detail solution procedure of implementing the IDEAL algorithm for compressible flows based on Delaunay triangular mesh is presented in this work, seemingly first in the literature.

Numerical study of binary mixture and thermophoretic analysis near a solar radiative heat transfer

The main focus of the current article is to investigate the two-dimensional boundary layer flow of non-Newtonian fluid induced by a linear stretching surface in the streamwise direction. The flow features of non-Newtonian fluid (i.e. shear-thinning and thickening nature of the fluids) are explored with Carreau fluid model. In addition, thermophoresis, Brownian motion, thermal radiation (with Rosseland approximation), binary mixture and activation enthalpy are also incorporated in fluid model for better analysis of thermal and solutal properties. The mathematical modelling of under consider physical problem yields nonlinear partial differential system. The governing mathematical system is first simplified with scaling transformations for the high Reynolds number and then transformed into non-dimensional ordinary differential system by using the similarity variables. Since the governing mathematical system comprised on nonlinear boundary value problem, thus shooting method is utilized for the numerical solution of the fluid flow governing equations. The obtained results followed the qualitative manners of the previously reported data. The computed results for important physical quantities (i.e. velocity, temperature, and concentration) are presented through graphs and then interpreted physically. It is observed that the boundary layer thickness increased for shear thickening fluids (n>1) while it is reduced for shear thinning fluids (n<1). Also, the temperature of shear thinning fluid is higher than shear thickening fluid. In addition, heat generation and thermal radiations enhance the thermal boundary layer. Thermophoretic viscosity effect reduces the concentration profile significantly in the flow domain.

Falkner–Skan boundary layer flow of a fluid with pressure-dependent viscosity past a stretching wedge with suction or injection

In this paper we study the boundary layer flow of a particular class of non-Newtonian fluids, namely piezo-viscous fluids, past a stretching wedge (Falkner–Skan flow). These fluids are essentially Navier–Stokes fluids with pressure-dependent viscosity. We consider the two-dimensional steady flow and introduce some similarity variables, so that the problem is reduced to a third-order nonlinear BVP. We solve the problem numerically by means of a shooting procedure combined with Newton’s method, and we find local similarity solutions. We investigate the behaviour of the velocity field, of the drag and of the thickness of the boundary layer for different values of the involved parameters, making a comparison between the piezo-viscous and the Newtonian case. We find that, depending on some material parameters, the effect of pressure is that of reducing or increasing the boundary layer with respect to the Newtonian case.

Thompson and Troian slip effect on ternary nanofluid flow over a stretching surface influenced by induced magnetic field and surface catalyzed reaction

This research scrutinizes the impact of the Thompson and Troian slip effect on a two-dimensional flow of ternary nanoliquid induced by an extending sheet near a stagnation point. Nanoparticles Iron oxide (Fe2O3), Nickle Zinc ferrite (NiZnFe2O4), and Magnesium zinc iron oxide (MnZnFe2O4) are dispersed in Ethylene glycol (C2H6O2) to establish ternary nanofluid. The induced magnetic field influences the flow, and the analysis of heat and mass transfers takes into account the Cattaneo–Christov (C–C) heat flux with homogenous–heterogeneous (H–H) reactions, respectively. To expedite the reaction, a surface-catalyzed process is introduced to efficiently control both (H–H) reactions within a reduced time frame. Utilizing the Tiwari and Das framework, the fluid flow characteristics are elucidated. Appropriate transformations are applied to derive ordinary differential equations (ODEs), which are then numerically deciphered via the bvp4c scheme. Graphs are portrayed to highlight the consequence of non-dimension quantities on the flow and temperature profiles. The surface drag coefficient and heat flux rate are also assessed and summarized. Outcomes revealed that for large estimates of the velocity slip parameter, the surface drag coefficient is enhanced while the heat flux rate is reduced. It is also interesting to note that because of the high estimations of ferromagnetic nanoparticles, the rate of heat transfer is elevated significantly. The validation of the proposed model is also included.

Similar Solutions for Changing Characteristics of Natural Convection Laminar Flow around a Vertical Rectangular Inclined Plane Surface

The study investigates laminar natural convection flow around a vertical rectangular inclined plane surface, specifically focusing on the effects of viscous and pressure stress work. Previous research primarily addressed these effects in two-dimensional natural convection flow scenarios, while this study extends the analysis to encompass three exterior situations, accounting for variations in fluid properties outside compressible boundary layers. Employing numerical methods, the study aims to predict essential flow parameters including Prandtl’s number (Pr), controlling parameter (C), and additional parameters (A, B, D, E, ∅). The investigation yields velocity profiles (F'(0), S'(0)), temperature profiles , skin frictions (F''(0), S''(0)), and heat transfer coefficients θ'(0). Numerical results are presented to illustrate the impact of different parameters on these flow characteristics. Additionally, tabulated data in Tables-2 and Tables-3 offer further insights into the predicted flow parameters and their variations under varying conditions.

Parametric analysis of the flow past a square cylinder in the interface of two different-velocity streams

We address the linear stability of the two-dimensional incompressible flow past a square cylinder immersed in the wake of an upstream splitter plate, which separates two streams of different velocities, UT (top) and UB (bottom), to three-dimensional perturbations. The analysis is done in the so-called wake transition regime, across which two-dimensional vortex shedding incorporates spanwise modulation. In addition to the top and bottom stream Reynolds numbers (ReT,B=DUT,B/ν, with D the square cylinder side and ν the kinematic viscosity of the fluid), a parametric analysis is conducted to gauge the effects of splitter plate length (S) and the gap between the trailing edge of the splitter plate and the front face of the cylinder (G) on the leading three-dimensionalizing instabilities. Modes akin to the A- and B-type instabilities that characterize the wake transition regime past circular and square cylinders in homogeneous incoming flow are observed only at very small values of the top-to-bottom Reynolds number ratio R=ReT/ReB, while a third mode, mode C, is ubiquitous beyond more than very mild values of R. Increasing S at constant small G has a stabilizing effect on mode C, whose onset is pushed to higher values of R. Only for long S is mode A observed. Fixing a short S and increasing G results instead in a destabilization of mode C, and mode B is favored over mode A.

Focal brain cooling suppresses spreading depolarization and reduces endothelial nitric oxide synthase expression in rats

This study aimed to investigate the effects of focal brain cooling (FBC) on spreading depolarization (SD), which is associated with several neurological disorders. Although it has been studied from various aspects, no medication has been developed that can effectively control SD. As FBC can reduce neuronal damage and promote functional recovery in pathological conditions such as epilepsy, cerebral ischemia, and traumatic brain injury, it may also potentially suppress the onset and progression of SD.We created an experimental rat model of SD by administering 1M potassium chloride (KCl) to the cortical surface. Changes in neuronal and vascular modalities were evaluated using multimodal recording, which simultaneously recorded brain temperature (BrT), wide range electrocorticogram, and two-dimensional cerebral blood flow. The rats were divided into two groups (cooling [CL] and non-cooling [NC]). Warm or cold saline was perfused on the surface of one hemisphere to maintain BrT at 37°C or 15°C in the NC and CL groups, respectively. Western blot analysis was performed to determine the effects of FBC on endothelial nitric oxide synthase (eNOS) expression.In the NC group, KCl administration triggered repetitive SDs (mean frequency = 11.57/h). In the CL group, FBC increased the duration of all KCl-induced events and gradually reduced their frequency. Additionally, eNOS expression decreased in the cooled brain regions compared to the non-cooled contralateral hemisphere.The results obtained by multimodal recording suggest that FBC suppresses SD and decreases eNOS expression. This study may contribute to developing new treatments for SD and related neurological disorders.

Two-dimensional shielded vortices in a shear current

The interaction of shielded vortices, with a continuous vorticity distribution, and a shear current of weak vorticity amplitude but similar velocity compared to the vortex amplitude is numerically investigated in two-dimensional isochoric flows. Different types of axisymmetric shielded vortices, namely, a neutral unstable vortex, a neutral robust vortex, and a non-neutral vortex are considered. The vortices are linear combinations of vorticity layer-modes, which consist of conveniently normalized cylindrical Bessel functions of order 0, truncated by a zero of the Bessel function of order 1. The vortex–current interaction is investigated by superposing initially the vortices at different initial locations along the cross-flow axis in the shear current. The numerical results show that some shielded vortices, as well as the shear current, remain robust while the vortices cross the shear current and reach a stable equilibrium location, which is of the same sign vorticity as its amount of circulation. There exist two unstable equilibrium locations where most of the vortices persist during a relatively short time interval before heading to their stable equilibrium region in the shear current.

Is a direct numerical simulation (DNS) of Navier-Stokes equations with small enough grid spacing and time-step definitely reliable/correct?

Turbulence is strongly associated with the vast majority of fluid flows in nature and industry. Traditionally, results given by the direct numerical simulation (DNS) of Navier-Stokes (NS) equations that relate to a famous millennium problem are widely regarded as ‘reliable’ benchmark solutions of turbulence, as long as grid spacing is fine enough (i.e. less than the minimum Kolmogorov scale) and time-step is small enough, say, satisfying the Courant-Friedrichs-Lewy condition (Courant number < 1). Is this really true? In this paper a two-dimensional sustained turbulent Kolmogorov flow driven by an external body force governed by the NS equations under an initial condition with a spatial symmetry is investigated numerically by the two numerical methods with detailed comparisons: one is the traditional DNS, the other is the ‘clean numerical simulation’ (CNS). In theory, the exact solution must have a kind of spatial symmetry since its initial condition is spatially symmetric. However, it is found that numerical noises of the DNS are quickly enlarged to the same level as the ‘true’ physical solution, which finally destroy the spatial symmetry of the flow field. In other words, the DNS results of the turbulent Kolmogorov flow governed by the NS equations are badly polluted mostly. On the contrary, the numerical noise of the CNS is much smaller than the ‘true’ physical solution of turbulence in a long enough interval of time so that the CNS result is very close to the ‘true’ physical solution and thus can remain symmetric, which can be used as a benchmark solution for comparison. Besides, it is found that numerical noise as a kind of artificial tiny disturbances can lead to huge deviations at large scale on the two-dimensional Kolmogorov turbulence governed by the NS equations, not only quantitatively (even in statistics) but also qualitatively (such as spatial symmetry of flow). This highly suggests that fine enough spatial grid spacing with small enough time-step alone could not guarantee the validity of the DNS of the NS equations: it is only a necessary condition but not sufficient. All of these findings might challenge some of our general beliefs in turbulence: for example, it might be wrong in physics to neglect the influences of small disturbances to NS equations. Our results suggest that, from physical point of view, it should be better to use the Landau-Lifshitz-Navier-Stokes (LLNS) equations, which consider the influence of unavoidable thermal fluctuations, instead of the NS equations, to model turbulent flows.

Robust and adaptive deep reinforcement learning for enhancing flow control around a square cylinder with varying Reynolds numbers

The present study applies a Deep Reinforcement Learning (DRL) algorithm to Active Flow Control (AFC) of a two-dimensional flow around a confined square cylinder. Specifically, the Soft Actor-Critic (SAC) algorithm is employed to modulate the flow of a pair of synthetic jets placed on the upper and lower surfaces of the confined squared cylinder in flow configurations characterized by Re of 100, 200, 300, and 400. The investigation starts with an analysis of the baseline flow in the absence of active control. It is observed that at Re = 100 and Re = 200, the vortex shedding exhibits mono-frequency characteristics. Conversely, at Re = 300 and Re = 400, the vortex shedding is dominated by multiple frequencies, which is indicative of more complex flow features. With the application of the SAC algorithm, we demonstrate the capability of DRL-based control in effectively suppressing vortex shedding, while significantly diminishing drag and fluctuations in lift. Quantitatively, the data-driven active control strategy results in a drag reduction of approximately 14.4%, 26.4%, 38.9%, and 47.0% for Re = 100, 200, 300, and 400, respectively. To understand the underlying control mechanism, we also present detailed flow field comparisons, which showcase the adaptability of DRL in devising distinct control strategies tailored to the dynamic conditions at varying Re. These findings substantiate the ability of DRL to control chaotic, multi-frequency dominated vortex shedding phenomena, underscoring the robustness of DRL in complex AFC problems.

A variational data assimilation approach for sparse velocity reference data in coarse RANS simulations through a corrective forcing term

The Reynolds-averaged Navier–Stokes (RANS) equations provide a computationally efficient method for solving fluid flow problems in engineering applications. However, the use of closure models to represent turbulence effects can reduce their accuracy. To address this issue, recent research has explored data-driven techniques such as data assimilation and machine learning.An efficient variational data assimilation (DA) approach is presented to enhance steady-state eddy viscosity based RANS simulations. To account for model deficiencies, a corrective force term is introduced in the momentum equation. In the case of only velocity reference data, this term can be represented by a potential field and is divergence-free. The DA implementation relies on the discrete adjoint method and approximations for efficient gradient evaluation.The implementation is based on a two-dimensional coupled RANS solver in OpenFOAM, which is extended to allow the computation of the adjoint velocity and pressure as well as the adjoint gradient. A gradient-based optimizer is used to minimize the difference between the simulation results and the reference data. To evaluate this approach, it is compared with alternative data assimilation methods for canonical stationary two-dimensional turbulent flow problems. For the data assimilation, sparsely distributed reference data from averaged high-fidelity simulation results are used.The results suggest that the proposed method achieves the optimization goal more efficiently compared to applying data assimilation for obtaining the eddy viscosity, or a field modifying the eddy viscosity, directly. The method works well for different reference data configurations and runs efficiently by leveraging coarse meshes.

Large-scale patterns set the predictability limit of extreme events in Kolmogorov flow

Events of extreme intensity in turbulent flows from atmospheric to industrial scales have a strong social and economic impact, and hence there is a need to develop models and indicators which enable their early prediction. Part of the difficulty here stems from the intrinsic sensitivity to initial conditions of turbulent flows. Despite recent progress in understanding and predicting extreme events, the question of how far in advance they can be ideally predicted (without model error and subject only to uncertainty in the initial conditions) remains open. Here we study the predictability limit of extreme dissipation bursts in the two-dimensional Kolmogorov flow by applying information-theoretic measures to massive statistical ensembles with more than $10^7$ direct numerical simulations. We find that extreme events with similar intensity and structure can exhibit disparate predictability due to different causal origins. Specifically, we show that highly predictable extreme events evolve from distinct large-scale circulation patterns. We thus suggest that understanding all the possible routes to the formation of extreme events is necessary to assess their predictability.