Unsteady Solutions Research Articles

Unsteady solute dispersion in large arteries under periodic body acceleration

The present study investigates the effect of periodic body acceleration on solute dispersion in blood flow through large arteries. Transport coefficients (i.e., exchange, convection, and dispersion coefficients) and mean concentration of the solute are analyzed in the presence of wall absorption. The solute is quickly transported to the wall of arteries with a smaller radius, whereas the opposite is true for arteries with a larger radius. In the presence of body acceleration, the amplitude of fluctuations of the convection coefficient K1(t) increases significantly as the radius of the artery increases. In contrast, an opposite scenario exists for the dispersion coefficient K2(t). The solute dispersion process becomes more effective in arterial blood flow as the radius of the artery decreases. More interestingly, in large arteries with body acceleration, the solute is convected, dispersed, and distributed more toward the upstream direction owing to flow reversal during the diastolic phase of pressure pulsation. Note that this important feature of flow reversal is solely due to periodic body acceleration. For an artery with a small radius, under the influence of periodic body acceleration, the mean concentration of solute Cm is the minimum, and more axial spread is noticed in the axial direction. In contrast, an opposite scenario arises in the artery with a large radius. Additionally, the effect of body acceleration on the shear-induced diffusion of red blood cells is discussed in blood flow.

Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems

We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak shock solution and the steady strong shock solution, when a steady supersonic flow impinges upon the solid wedge – the half-angle of which is less than a critical angle (i.e., the detachment angle), and then conjectured that the steady weak shock solution is physically admissible since it is the one observed experimentally. The fundamental issue of whether one or both of the steady weak and strong shocks are physically admissible has been vigorously debated over the past eight decades and has not yet been settled in a definitive manner. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the two-dimensional Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions of multidimensional hyperbolic systems of conservation laws. In this sense, we have to understand the reflection configurations in order to understand fully the global entropy solutions of two-dimensional hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory for all the physical parameters up to the detachment angle.

Unsteady solute dispersion of electro-osmotic flow of micropolar fluid in a rectangular microchannel

This study scrutinizes the two-dimensional concentration distribution for a solute cloud containing a micropolar fluid in a rectangular microchannel under the influence of an applied electric field. The concentration distribution is obtained up to second order approximation using Mei's homogenization method. Analytical formulas are derived for dispersion coefficient, mean and two-dimensional concentration distributions. This study also includes the analytical expressions for electric potential, velocity, and microrotation profiles. This study discusses the impact of coupling number, couple stress parameter, electric double layer thickness, and Péclet number on solute concentration distribution. The results of fluid velocity and dispersion coefficient are validated with available works in the literature. The non-Newtonian parameter and electric double layer thickness are shown to have a significant impact on dispersion. Our study reveals that concentration distribution rises but spreading of solute reduces when the coupling number increases. This is also true when the Debye length decreases. It is also obtained that the solute spreads more in the Newtonian fluid case compared to the micropolar fluid case. Finally, coupling number and electric double layer thickness show a symmetric pattern to the indicator function for the transverse concentration variation rate. The findings of this work have broad implications in deoxyribonucleic acid analysis, chemical mixing, and separation.

Aeroacoustics computation for propellers based on harmonic balance solution

In this paper, a novel open-source framework is presented for the evaluation of noise emissions produced by aerodynamic bodies exhibiting dominant motion at specific frequencies. This behavior is frequently encountered in propellers used for urban air mobility applications. The reduced order model called harmonic balance is employed to compute the unsteady flow solution, reducing the computational cost. The approach tackles K different frequencies by capturing the flow solution at N discrete time instances within a single period, where N is defined as 2K+1. The time history of the conservative variables on the solid surfaces is reconstructed through a Fourier interpolation. The Kirchhoff Ffowcs Williams Hawkings integral formulation, integrated into SU2, is used to compute the sound pressure level perceived by farfield observers. The integral formulation propagates the acoustic solution with a computational cost independent of observer distance. The noise emission of a pitching wing and a propeller is computed with the proposed framework. Regarding a small, isolated propeller operating at low Reynolds numbers in forward flight, we conducted a comparison of the aerodynamic and aeroacoustic results between a fully time-accurate solution and a steady-state Reynolds Averaged Navier-Stokes solution within a rotating reference frame. The harmonic balance results demonstrated improved agreement with the fully unsteady solution across the first three blade-pass frequencies and exhibited this consistency over a broader range of propagation angles.

A modified lattice Boltzmann approach based on radial basis function approximation for the non‐uniform rectangular mesh

AbstractWe have presented a novel lattice Boltzmann approach for the non‐uniform rectangular mesh based on the radial basis function approximation (RBF‐LBM). The non‐uniform rectangular mesh is a good option for local grid refinement, especially for the wall boundaries and flow areas with intensive change of flow quantities. Which allows, the total number of grid cells to be reduced and so the computational cost, therefore improving the computational efficiency. But the grid structure of the non‐uniform rectangular mesh is no longer applicable to the classic lattice Boltzmann method (CLBM), which is based on the famous BGK collision‐streaming evolution. This is why the present study is inspired by the idea of the interpolation‐supplemented LBM (ISLBM) methodology. The ISLBM algorithm is improved in the present manuscript and developed into a novel LBM approach through the radial basis function approximation instead of the Lagrangian interpolation scheme. The new approach is validated for both steady states and unsteady periodic solutions. The comparison between the radial basis function approximation and the Lagrangian interpolation is discussed. It is found that the novel approach has a good performance on computational accuracy and efficiency. Proving that the non‐uniform rectangular mesh allows grid refinement while obtaining precise flow predictions.

Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior – an approximation of the steady solution – which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

A multi-region analysis of unsteady Oseen's equation for accelerating flow past a circular cylinder

It is difficult to find a valid high-order (of Reynolds number) analytical approximation to the uniform viscous flow past an infinite circular cylinder by applying perturbation techniques. In this study, we show that by accelerating flow past a circular cylinder with taking unsteady Stokes' solution as an initial approximation, higher-order approximation solutions to unsteady Oseen's equation can be obtained by an iteration scheme of perturbation, which exactly satisfy the boundary conditions. As a matter of fact, the nonlinear (convective) term linearized by Oseen's approximation is overweighted especially near the body surface. To eliminate the overweight, a multi-region analysis is proposed in this study to improve analytical unsteady Oseen's solution so as to extend the valid range of Reynolds numbers. In other words, the flow region is hypothetically divided into several annular regions, and the linearized convective term in each region is modified by multiplying with a Carrier's coefficient c (0 < c ≤ 1). The results show that, when the accelerating parameter in the range of 0.5 ≤ a ≤ 4, the maximum effective Reynolds number Reeff of the fourth-order five-region solution is both much larger than that of Stokes' and Oseen's ones. For example, when a = 0.5, Reeff = 22.26 is roughly eight times that of Stokes' solution (Reeff = 2.67), and nine times that of Oseen's solution (Reeff = 2.41). Moreover, when the instantaneous Reynolds number Re(t) is smaller than the Reeff, the flow separation angle and the wake length are both consistent with the numerical results obtained by accurately solving the full Navier-Stokes equations. In addition, the flow properties, including the drag coefficients, streamline patterns, and the pressure coefficients, as well as the vorticity distributions also agree well with the numerical results. This study represents a significant extension of our previous study for unsteady Stokes' equation [Xu et al., Phys. Fluids 35, 033608 (2023)] to unsteady Oseen's equation and its generalization.

Optimization of suspended particulate transport parameters from measured concentration profiles with a new analytical model

The water body’s suspended concentration reflects many coastal environmental indicators, which is important for predicting ecological hazards. The modeling of any concentration in water requires solving the settling-diffusion equation (SDE), and the values of several key input parameters therein (settling velocity ws, eddy diffusivity Ds, and erosion rates p(t)) directly determine the prediction performance. The time-consuming large-scale simulations would benefit if the parameter values could be estimated through available observations in the target sea area. The present work proposes a new optimization method for synchronously estimating the three parameters from limited concentration observations. First, an analytical solution to the one-dimensional vertical (1DV) SDE for suspended concentrations in an unsteady scenario is derived. Second, the near bottom suspended sediment concentration (SSC) profiles are measured with high-resolution observation. Third, the key parameters are optimized through the best fit of the measured SSC profiles and those modeled with the unsteady solution. Nonlinear least square fitting (NLSF) is introduced to judge the best fits automatically. The high-resolution concentration measurements in a specially-designed cylindrical tank experiment using the Yellow River Delta sediments test the proposed method. The method performs well in the initial period of turbulence generation when sediment resuspension is significant. It optimizes p(t), ws, and Ds with reasonable values and uniqueness of their combination. The proposed theory is a practical tool for quickly estimating key substance transport parameters from limited observations; it also has the potential to construct local parametric models to benefit the 3D modeling of coastal substance transport. Although the present work takes SSC as an example, it can be extended to any suspended particulate concentration in the water.

Unsteady solute transport in Casson fluid flow and its retention in an atherosclerotic wall

In the current study, a mathematical model of convective solute transport and its retention in an atherosclerotic wall is examined. The artery wall is supposed to be permeable and has cosine-shaped stenosis caused by various sorts of abnormal growth or plaque formation. The flowing blood is modeled as the suspension of all red blood cells in plasma, thought to be Casson fluid. The Immersed Boundary Method (IBM) and the Marker and Cell (MAC) method are used to numerically solve the unsteady governing equations of motion along with the appropriate initial and boundary conditions. The final results of the quantitative studies contain the corresponding flow-field and solute distribution profiles for the full arterial length. The essential components are evaluated, including the wall shear stress, wall concentration, and concentration contours. According to simulation data, the permeability of the wall directly affects the solute transport within the therapeutic region. The yield stress does, however, increase flow resistance and decrease solute uptake within the tissue. The projected outcomes are remarkably in line with previously published findings, which supports the applicability of the model under investigation.

Modal analysis for incompressible fluid flow

This paper presents a numerical method for incompressible fluid flow. A difficulty in analyzing incompressible fluid flow is that the continuity equation has no time evolution term. In the marker and cell (MAC) method, Poisson’s equation is solved iteratively, which takes most of the computation time, and in the artificial compressibility method (ACM), pseudo-time iteration is necessary to solve for unsteady solutions. Here, modal analysis that uses the velocity eigenvectors corresponding to zero eigenvalues is proposed for analyzing two-dimensional incompressible fluid flow. The proposed method involves only about one third of the number of variables needed in the MAC method and the ACM, and it does not require iterative calculation of Poisson’s equation or pseudo-time iteration. Numerical results for a simple flow system and a cavity flow obtained using the proposed method are compared with those obtained using the ACM and the simplified MAC method. The results agree well, thereby validating the proposed modal analysis.

Gradient Decay in the Boltzmann Theory of Non-isothermal Boundary

We consider the Boltzmann equation in a convex domain with a non-isothermal boundary of diffuse reflection. For both unsteady/steady problems, we construct solutions belonging to Wx1,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{1,p}_x$$\\end{document} for any p<3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p<3$$\\end{document}. We prove that the unsteady solution converges to the steady solution in the same Sobolev space exponentially quickly as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\rightarrow \\infty $$\\end{document}.

Cooling enhancement for engine parts using jet impingement

A computational study has been performed to evaluate the use of jet impingement for cooling applications in the automotive industry. The current study uses an entire internal combustion engine cylinder with its components as a computational domain. An unsteady numerical solution for the Navier-Stokes equations was carried out using Improved Delayed Detached Eddy Simulation (IDDES). The volume of fluid approach is proposed to track and locate the liquid jet surface that is in contact with the air. The conjugate heat transfer approach is used to link the heat transfer solution between the fluid and the solid. The boundary conditions that are employed in the study are provided from lab experiments and one-dimensional simulations. The cooling jet in this study targets the hottest region in the piston, i.e., the region underneath the exhaust valve. Three nozzle sizes with flows at different Reynolds numbers are chosen to examine the thermal characteristics of the cooling jet. The computational study reveals that for a specific Reynolds number, the smaller diameter nozzle provides the highest heat transfer coefficient around the impingement point. The maximum relative velocity location at the impingement point slightly leads the location of the maximum Nusselt number. The maximum temperature in the piston decreases by 7% to 11% as the nozzle diameter changes from 1.0 to 3.0 mm for a jet Reynolds number of 4,500. If a correct selection is made for the nozzle size, the cooling jet can be efficiently used to reduce the temperature and alleviate the thermal stresses in the piston in the region underneath the exhaust valve where the maximum temperature occurs.

Multiple bubble dynamics and velocity selection in Laplacian growth without surface tension

A new selection phenomenon in nonlinear interface dynamics is predicted. A generic class of exact regular unsteady multi-bubble solutions in a Hele-Shaw cell is presented. These solutions show that the case where the asymptotic bubble velocity, U, is twice greater than the velocity, V, of the uniform background flow, i.e., U=2V, is the only attractor of the dynamics. Contrary to common belief, the predicted velocity selection requires neither surface tension nor other external regularization.

A general unsteady Fourier solution for orthotropic heat transfer in 2D functionally graded cylinders

In the present research, an analytical solution is presented for unsteady heat transfer in a two‐dimensional finite cylinder made of functionally graded Materials (FGM). The governing equation, as an unsteady partial differential equation (PDE), is solved with general boundary conditions (BCs). The thermal conductivity coefficients are considered as a power function of the radius, both in the longitudinal and radial directions of the FGM cylinder. The presence of non‐constant coefficients and general BCs will confront us with the challenge of solving an unsteady problem with high complexity. A combination of Laplace transform, Fourier transform, and meromorphic functions are utilized to deal with this complex PDE. The Laplace transform was applied to transform to the frequency domain, the Sturm–Liouville equation was used to extract the orthogonal basis Fourier series, and the meromorphic function was deployed to transform back from the frequency domain to the time domain. Results were well verified against previous research works. Results showed that the proposed analytical solution could well predict the temperature distribution. The current analytical solution is useful to better understand unsteady heat transfer in FGMs.

A computational modeling on two-dimensional laminar flow and thermal characteristics through a strongly bent square channel

In order to have a precise knowledge on how pressure gradients and buoyancy force affect fluid flow and energy distribution in a bending channel, it is important to perform a comprehensive study on flow characteristics and heat transfer mechanisms that trigger out the transition of fluids into a turbulent state, subject to a sustained pressure gradient. The present paper explores a computational modeling on two-dimensional fluid flow and thermal characteristics in a bent square channel of strong curvature. The Newton–Raphson (N-R) iteration method is applied to obtain a bifurcation structure depending on the pressure-driven force, the Dean number (De), covering 0 &amp;lt; De ≤ 5000. As a consequence, four branches of asymmetric steady solutions are identified for each of the cases of the Grashof number, Gn (=1000, 1500, and 2000), where only the first branch is found to exhibit asymmetric two-vortex solutions while the remaining branches encompass two- to four-vortex solutions. The similarity and disparity in the branching structure are also demonstrated. Then, adopting the Adam–Bashforth (A-B) method together with Crank–Nicholson (C-N) formula, the unsteady solutions (US) have been explored, validated by power spectrum density (PSD) and phase space Within the realm of US, two- and three-vortex solutions are found and these solutions exhibit transitions from steady to chaotic behavior profoundly. Effects of the Grashof number with convective heat transfer (CHT) are also compared. By analyzing the Nusselt number (Nu), it is observed that in case of highly chaotic flow, CHT experiences substantial enhancement. This intensified CHT arises from increased turbulence and mixing, facilitating more efficient thermal energy exchange under such chaotic flow conditions.

Numerical study of solution structure and nonlinear behavior of Dean flow with vortex structure in a bending square duct

Fluid flow and heat transfer in bending channels are topics of much research interest because of increasing demand in various fields, particularly in the medical and industrial arenas. This paper presents a numerical study of fluid flow and heat transfer in a bending channel with a square cross section. Numerical calculations using spectral methods were performed for a curvature of 0.001 and a Dean number (Dn) in the range of 0&amp;lt;Dn≤5000. A temperature difference was maintained between the horizontal walls for a Grashof number of Gr = 1000, with the bottom wall heated and the outer and inner walls thermally insulated. Applying Newton–Raphson iteration and path continuation, two branching structures of steady solutions with two to eight vortices were obtained. The first branch comprises multi-vortex up to eight and it occurs at Dn≥3500 while the second branch comprises to multi-vortex up to a maximum of four. The first branch forms symmetric solution while the second branch for its complexity forms asymmetric solutions. For unsteady solutions, time-evolution calculations were performed to investigate the nonlinear behavior, and it was found that with increasing Dn, the unsteady flow experienced various flow instabilities. The time evolution was plotted in phase space to clarify the unsteady flow characteristics. Distinctive contours of the secondary flow patterns, streamwise velocity distributions, and isotherms were also obtained, and the unsteady flow was found to comprise two to six vortices. Finally, convective heat transfer was explored by obtaining temperature contours, and the secondary flow was found to magnify the convective heat transfer significantly. Because of the increase of several secondary vortices in the chaotic solutions, heat transfer occurred markedly in the flow.

Rimming flow in a rotating horizontal cylinder with phase change at the interface

The two-dimensional solutions and stability analysis are presented for an evaporating thin viscous liquid film flowing inside a uniformly heated rotating horizontal cylinder. A non-linear, fourth-order, partial differential evolution equation is obtained by simplifying mass, momentum, and energy conservation equations within the lubrication approximation. The effect of evaporation, gravity, viscous drag, surface tension, thermocapillary stress, and intermolecular forces has been taken into account. The numerical solutions of the model are validated against the existing experimental as well as the numerical results, along with the analytical result for the limiting cases of the present model. The film thickness model is solved to elucidate two-dimensional spatiotemporal solutions and their stability for a wide range of thermal and other parameters. The evaporative mass flux at the liquid–air interface results in unsteady solutions which are oscillatory in nature, and the amplitude of the oscillations increases with an increase in the evaporative flux. The film ruptures after some time and the rupture time is found to be inversely proportional to the evaporation number, a non-dimensional number quantifying the rate of evaporation. The linear stability analysis shows that the thermocapillary stresses as well as the long-range molecular forces destabilize the film. A negative disjoining pressure is shown to reduce the rupture time and vice versa. Evaporation (condensation) plays a destabilizing (stabilizing) role in the thin film flow. Non-linear computations are carried out for the steady profiles, validating the growth rates obtained from the linear stability analysis.

Exact Analysis of Unsteady Solute Dispersion in Blood Flow: A Theoretical Study

The diameter of an artery can narrow due to atherosclerosis or stenosis, making it challenging to resolve solute dispersion issues as blood flows via a stenosed artery. The stenosis occurrence restricted drug dispersion and blood flow. This research introduces the establishment of a mathematical model in examining the unsteady dispersion with respect to the solute in overlapping stenosis arteries depicting blood as a Herschel-Bulkley (H-B) fluid model. Note that fluid velocity was obtained by analytically solving the governing and constitutive equations. The transport equation has been solved by employing a generalised dispersion model (GDM), in which the dispersion process is described. Accordingly, yield stress, stenosis height, slug input of solute length, as well as a rise in the power-law index have improved the peak with regard to the mean concentration and solute concentration. The maximum mean concentration yielded the effective dose for therapeutic concentration. In conclusion, this study is relevant to disease arteries, coagulating hemodynamics and may help physiologists in furnishing a more refined understanding of diffusion processes in cardiovascular hydrodynamics. An interesting application related to the present study is the transportation of drugs in the arterial blood flow.

Three-dimensional unsteady state solution of multilayered saturated anisotropic finite media subjected to point source

The three-dimensional (3D) unsteady state solution of a saturated anisotropic finite porous medium due to a point source is presented in this paper. First, Laplace transform is introduced to address the governing equations of anisotropic poroelasticity. Next, the eigen equation method and the modified pseudo-Stroh formalism are used to obtain the transient solution for homogeneous saturated finite media. Then, the propagation matrix method is used to address the corresponding multilayered porous elastic media, and the 3D unsteady state solution of the layered porous medium subjected to a point source is obtained. The solution for a finite porous foundation is presented to demonstrate the use of the closed form solution. The numerical results of a homogeneous foundation due to an internal time-dependent point source are given. Two different point sources are used to show the transient responses of the proposed method. By comparing with those obtained by the finite element method, the accuracy of the proposed method has been proved. Meanwhile, the influence of the depth of the point source on the hydromechanical responses of the homogeneous foundation is discussed. Finally, the transient response of an anisotropic sandwich foundation due to a point source at different depths is investigated. These numerical examples prove the feasibility of the transient 3D solution. The closed form solution presented in this paper can provide help for 3D transient fluid-deforming coupling analysis of laminated porous media.

On the Nature of Transonic Shock Buffet in an Axial-Flow Fan

Transonic buffet or self-sustained shock oscillation in transonic flow is not well-understood in fans and compressors. The present work is a numerical study of transonic buffet flow using a full-annulus model of a transonic fan—the NASA rotor 67. Firstly, results obtained with steady simulations are validated with experimental and numerical data available, and shock structures from these simulations are found to exhibit distinct features as seen in experiments. Unsteady Reynolds-averaged Navier–Stokes simulations are then carried out at two operating points—near design mass flow rate and toward stall—on the fan characteristic to capture transonic shock buffet. The unsteady simulations are able to capture necessary attributes of transonic buffet flow. Further, wave propagation in an unsteady flow solution featuring transonic buffet is investigated to understand the mechanism driving shock buffet in axial turbomachines. Upstream- and downstream-propagating pressure perturbation waves are observed on the blade suction surface within a buffet cycle, while only an upstream-propagating wave is seen on the blade pressure surface. The driving mechanism of transonic buffet in this turbomachine is elucidated through a feedback loop of the upstream- and downstream-propagating waves—qualitatively and quantitatively. Besides the streamwise pressure waves, circumferential pressure waves propagating at part speed of the rotor in the spinning direction and radially outward moving pressure waves of similar speeds on the blade suction and pressure surfaces are also detected as distinct features of buffet flow for this turbomachine.